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Working in the language of Ackermann set theory:

Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\vec{x}$ are the parameters.

Let $\phi^*(\vec{P^*},\vec{x})$ be the formula obtained from $\phi(\vec{P},\vec{x})$ by replacing each atomic formula $P_i(y_1,..,y_k)$ by $ \langle y_1,..,y_k \rangle \in P^*_i$, where each $P^*_i$ is a variable symbol.

Where: $\langle x \rangle = x \\ \langle x_1,x_2 \rangle=\{\{x_1\},\{x_1,x_2\}\} \\\langle x_1,..,x_{n+1} \rangle = \langle \langle x_1,..,x_n \rangle, x_{n+1} \rangle $

If we add the following rule:

$ \forall \vec{x} \in V \, \exists x \in V \, \forall y \ (y \in x \iff \phi(\vec{P},\vec{x})) \\... \\ ... \\ \overline { \forall \vec{P^*} \subseteq V \, \forall \vec{x} \in V\, \exists x \in V \, \forall y \ (y \in x \iff \phi^*(\vec{P^*},\vec{x}))} $

to the axioms of $\sf Ackermann - Second \ completeness \ axiom \ for \ V $.

[Note: the input of the above rule is a schema, where we have a single sentence per each substitution of the predicate symbols $\vec{P}$ in $\phi$. For example if $\phi(P,x_1,y)$ is $y \in x_1 \land P(y)$, then we'll have a sentence per each substituion of $P$ by a set theoretic definable predicate].

What would be the strength of this theory?

The idea is that $\sf Ackermann - \text {Second completeness axiom for } V$ is equivalent to $\sf Z_2$, now it proves set theoretic separation for sets, but with the above rule, this would turn to prove the second completeness axiom for $V$ (i.e. subclasses of sets are sets), and this would blow up that theory to the level of $\sf ZFC$, a great jump! Now, similarly replacement schema would be proved for set theoretic formulas closed on sets, but with this rule added it would lead to proving limitation of size axiom, which will blow up the consistency strength of this theory up to near Mahlo cardinals.

So, where do the jump in consistency strength stop?

A wilder stretch is to have the rule unrestricted for each bounded $\in V$ sentence $\psi^V(\vec{P})$, so formally:

$\psi^V(\vec{P}) \\ ... \\... \\\overline {\forall P^* \subseteq V \, (\psi^V(\vec{P^*}))}$

Working in the language of Ackermann set theory:

Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\vec{x}$ are the parameters.

Let $\phi^*(\vec{P^*},\vec{x})$ be the formula obtained from $\phi(\vec{P},\vec{x})$ by replacing each atomic formula $P_i(y_1,..,y_k)$ by $ \langle y_1,..,y_k \rangle \in P^*_i$, where each $P^*_i$ is a variable symbol.

Where: $\langle x \rangle = x \\ \langle x_1,x_2 \rangle=\{\{x_1\},\{x_1,x_2\}\} \\\langle x_1,..,x_{n+1} \rangle = \langle \langle x_1,..,x_n \rangle, x_{n+1} \rangle $

If we add the following rule:

$ \forall \vec{x} \in V \, \exists x \in V \, \forall y \ (y \in x \iff \phi(\vec{P},\vec{x})) \\... \\ ... \\ \overline { \forall \vec{P^*} \subseteq V \, \forall \vec{x} \in V\, \exists x \in V \, \forall y \ (y \in x \iff \phi^*(\vec{P^*},\vec{x}))} $

to the axioms of $\sf Ackermann - Second \ completeness \ axiom \ for \ V $.

[Note: the input of the above rule is a schema, where we have a single sentence per each substitution of the predicate symbols $\vec{P}$ in $\phi$. For example if $\phi(P,x_1,y)$ is $y \in x_1 \land P(y)$, then we'll have a sentence per each substituion of $P$ by a set theoretic definable predicate].

What would be the strength of this theory?

The idea is that $\sf Ackermann - \text {Second completeness axiom for } V$ is equivalent to $\sf Z_2$, now it proves set theoretic separation for sets, but with the above rule, this would turn to prove the second completeness axiom for $V$ (i.e. subclasses of sets are sets), and this would blow up that theory to the level of $\sf ZFC$, a great jump! Now, similarly replacement schema would be proved for set theoretic formulas closed on sets, but with this rule added it would lead to proving limitation of size axiom, which will blow up the consistency strength of this theory up to near Mahlo cardinals.

So, where do the jump in consistency strength stop?

A wilder stretch is to have the rule unrestricted for each bounded $\in V$ sentence $\psi^V(\vec{P})$ [where $V$ doesn't occur other than in bounding quantifiers], so formally:

$\psi^V(\vec{P}) \\ ... \\... \\\overline {\forall P^* \subseteq V \, (\psi^V(\vec{P^*}))}$

Working in the language of Ackermann set theory:

Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\vec{x}$ are the parameters.

Let $\phi^*(\vec{P^*},\vec{x})$ be the formula obtained from $\phi(\vec{P},\vec{x})$ by replacing each atomic formula $P_i(y_1,..,y_k)$ by $ \langle y_1,..,y_k \rangle \in P^*_i$, where each $P^*_i$ is a variable symbol.

Where: $\langle x \rangle = x \\ \langle x_1,x_2 \rangle=\{\{x_1\},\{x_1,x_2\}\} \\\langle x_1,..,x_{n+1} \rangle = \langle \langle x_1,..,x_n \rangle, x_{n+1} \rangle $

If we add the following rule:

$ \forall \vec{x} \in V \, \exists x \in V \, \forall y \ (y \in x \iff \phi(\vec{P},\vec{x})) \\... \\ ... \\ \overline { \forall \vec{P^*} \subseteq V \, \forall \vec{x} \in V\, \exists x \in V \, \forall y \ (y \in x \iff \phi^*(\vec{P^*},\vec{x}))} $

to the axioms of $\sf Ackermann - Second \ completeness \ axiom \ for \ V $.

[Note: the input of the above rule is a schema, where we have a single sentence per each substitution of the predicate symbols $\vec{P}$ in $\phi$. For example if $\phi(P,x_1,y)$ is $y \in x_1 \land P(y)$, then we'll have a sentence per each substituion of $P$ by a set theoretic definable predicate].

What would be the strength of this theory?

The idea is that $\sf Ackermann - \text {Second completeness axiom for } V$ is equivalent to $\sf Z_2$, now it proves set theoretic separation for sets, but with the above rule, this would turn to prove the second completeness axiom for $V$ (i.e. subclasses of sets are sets), and this would blow up that theory to the level of $\sf ZFC$, a great jump! Now, similarly replacement schema would be proved for set theoretic formulas closed on sets, but with this rule added it would lead to proving limitation of size axiom, which will blow up the consistency strength of this theory up to near Mahlo cardinals.

So, where do the jump in consistency strength stop?

A wilder stretch is to have the rule unrestricted for each sentence $\psi(\vec{P})$, so formally:

$\psi(\vec{P}) \\ ... \\... \\\overline {\forall P^* \subseteq V \, (\psi(\vec{P^*}))}$

Working in the language of Ackermann set theory:

Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\vec{x}$ are the parameters.

Let $\phi^*(\vec{P^*},\vec{x})$ be the formula obtained from $\phi(\vec{P},\vec{x})$ by replacing each atomic formula $P_i(y_1,..,y_k)$ by $ \langle y_1,..,y_k \rangle \in P^*_i$, where each $P^*_i$ is a variable symbol.

Where: $\langle x \rangle = x \\ \langle x_1,x_2 \rangle=\{\{x_1\},\{x_1,x_2\}\} \\\langle x_1,..,x_{n+1} \rangle = \langle \langle x_1,..,x_n \rangle, x_{n+1} \rangle $

If we add the following rule:

$ \forall \vec{x} \in V \, \exists x \in V \, \forall y \ (y \in x \iff \phi(\vec{P},\vec{x})) \\... \\ ... \\ \overline { \forall \vec{P^*} \subseteq V \, \forall \vec{x} \in V\, \exists x \in V \, \forall y \ (y \in x \iff \phi^*(\vec{P^*},\vec{x}))} $

to the axioms of $\sf Ackermann - Second \ completeness \ axiom \ for \ V $.

[Note: the input of the above rule is a schema, where we have a single sentence per each substitution of the predicate symbols $\vec{P}$ in $\phi$. For example if $\phi(P,x_1,y)$ is $y \in x_1 \land P(y)$, then we'll have a sentence per each substituion of $P$ by a set theoretic definable predicate].

What would be the strength of this theory?

The idea is that $\sf Ackermann - \text {Second completeness axiom for } V$ is equivalent to $\sf Z_2$, now it proves set theoretic separation for sets, but with the above rule, this would turn to prove the second completeness axiom for $V$ (i.e. subclasses of sets are sets), and this would blow up that theory to the level of $\sf ZFC$, a great jump! Now, similarly replacement schema would be proved for set theoretic formulas closed on sets, but with this rule added it would lead to proving limitation of size axiom, which will blow up the consistency strength of this theory up to near Mahlo cardinals.

So, where do the jump in consistency strength stop?

A wilder stretch is to have the rule unrestricted for each bounded $\in V$ sentence $\psi^V(\vec{P})$, so formally:

$\psi^V(\vec{P}) \\ ... \\... \\\overline {\forall P^* \subseteq V \, (\psi^V(\vec{P^*}))}$

Working in the language of Ackermann set theory:

Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\vec{x}$ are the parameters.

Let $\phi^*(\vec{P^*},\vec{x})$ be the formula obtained from $\phi(\vec{P},\vec{x})$ by replacing each atomic formula $P_i(y_1,..,y_k)$ by $ \langle y_1,..,y_k \rangle \in P^*_i$, where each $P^*_i$ is a variable symbol.

Where: $\langle x \rangle = x \\ \langle x_1,x_2 \rangle=\{\{x_1\},\{x_1,x_2\}\} \\\langle x_1,..,x_{n+1} \rangle = \langle \langle x_1,..,x_n \rangle, x_{n+1} \rangle $

If we add the following rule:

$ \forall \vec{x} \in V \, \exists x \in V \, \forall y \ (y \in x \iff \phi(\vec{P},\vec{x})) \\... \\ ... \\ \overline { \forall \vec{P^*} \subseteq V \, \forall \vec{x} \in V\, \exists x \in V \, \forall y \ (y \in x \iff \phi^*(\vec{P^*},\vec{x}))} $

to the axioms of $\sf Ackermann - Second \ completeness \ axiom \ for \ V $.

[Note: the input of the above rule is a schema, where we have a single sentence per each substitution of the predicate symbols $\vec{P}$ in $\phi$. For example if $\phi(P,x_1,y)$ is $y \in x_1 \land P(y)$, then we'll have a sentence per each substituion of $P$ by a set theoretic definable predicate].

What would be the strength of this theory?

The idea is that $\sf Ackermann - \text {Second completeness axiom for } V$ is equivalent to $\sf Z_2$, now it proves set theoretic separation for sets, but with the above rule, this would turn to prove the second completeness axiom for $V$ (i.e. subclasses of sets are sets), and this would blow up that theory to the level of $\sf ZFC$, a great jump! Now, similarly replacement schema would be proved for set theoretic formulas closed on sets, but with this rule added it would lead to proving limitation of size axiom, which will blow up the consistency strength of this theory up to near Mahlo cardinals.

So, where do the jump in consistency strength stop?

Working in the language of Ackermann set theory:

Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\vec{x}$ are the parameters.

Let $\phi^*(\vec{P^*},\vec{x})$ be the formula obtained from $\phi(\vec{P},\vec{x})$ by replacing each atomic formula $P_i(y_1,..,y_k)$ by $ \langle y_1,..,y_k \rangle \in P^*_i$, where each $P^*_i$ is a variable symbol.

Where: $\langle x \rangle = x \\ \langle x_1,x_2 \rangle=\{\{x_1\},\{x_1,x_2\}\} \\\langle x_1,..,x_{n+1} \rangle = \langle \langle x_1,..,x_n \rangle, x_{n+1} \rangle $

If we add the following rule:

$ \forall \vec{x} \in V \, \exists x \in V \, \forall y \ (y \in x \iff \phi(\vec{P},\vec{x})) \\... \\ ... \\ \overline { \forall \vec{P^*} \subseteq V \, \forall \vec{x} \in V\, \exists x \in V \, \forall y \ (y \in x \iff \phi^*(\vec{P^*},\vec{x}))} $

to the axioms of $\sf Ackermann - Second \ completeness \ axiom \ for \ V $.

[Note: the input of the above rule is a schema, where we have a single sentence per each substitution of the predicate symbols $\vec{P}$ in $\phi$. For example if $\phi(P,x_1,y)$ is $y \in x_1 \land P(y)$, then we'll have a sentence per each substituion of $P$ by a set theoretic definable predicate].

What would be the strength of this theory?

The idea is that $\sf Ackermann - \text {Second completeness axiom for } V$ is equivalent to $\sf Z_2$, now it proves set theoretic separation for sets, but with the above rule, this would turn to prove the second completeness axiom for $V$ (i.e. subclasses of sets are sets), and this would blow up that theory to the level of $\sf ZFC$, a great jump! Now, similarly replacement schema would be proved for set theoretic formulas closed on sets, but with this rule added it would lead to proving limitation of size axiom, which will blow up the consistency strength of this theory up to near Mahlo cardinals.

So, where do the jump in consistency strength stop?

A wilder stretch is to have the rule unrestricted for each sentence $\psi(\vec{P})$, so formally:

$\psi(\vec{P}) \\ ... \\... \\\overline {\forall P^* \subseteq V \, (\psi(\vec{P^*}))}$