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Statement 1 is not directly expressible in ZFC or GBC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.

But if you have ZFC in the language with a class truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is both expressible and provable in the theory GBC+$T$ is a first-order truth predicate.

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different 1-types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same 1-type, that is, with the same pattern of truths for all the various formulas $\varphi$. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.

Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any substructure of $W$ of size larger than $|\sigma|$, and by Löwenheim-Skolem, let $X$ be any elementary substructure of $Y$ of size $|\sigma|$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.

Statement 1 is very nice, it says that in any proper class, there are two distinct objects $X$ and $Y$ that are indiscernible with respect to assertions in the ambiant theory (I assume the language of set theory); they exhibit exactly the same pattern of truths: $\varphi(X)$ if and only if $\varphi(Y)$ for every assertion $\varphi$.

This assertion is not directly expressible in ZFC or GBC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.

But if you do have a truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is both expressible and provable in the theory GBC + $T$ is a first-order truth predicate.

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different 1-types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same 1-type, that is, with the same pattern of truths for all the various formulas $\varphi$. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.

Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any substructure of $W$ of size larger than $|\sigma|$, and by Löwenheim-Skolem, let $X$ be any elementary substructure of $Y$ of size $|\sigma|$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.

Statement 1 is not directly expressible in ZFC or GBC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.

But if you have ZFC in the language with a class truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is both expressible and provable in the theory GBC+$T$ is a first-order truth predicate.

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same type. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.

Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any uncountable substructure of $W$, and by Löwenheim-Skolem, let $X$ be any countable elementary substructure of $Y$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.

Statement 1 is not directly expressible in ZFC or GBC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.

But if you have ZFC in the language with a class truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is both expressible and provable in the theory GBC+$T$ is a first-order truth predicate.

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different 1-types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same 1-type, that is, with the same pattern of truths for all the various formulas $\varphi$. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.

Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any substructure of $W$ of size larger than $|\sigma|$, and by Löwenheim-Skolem, let $X$ be any elementary substructure of $Y$ of size $|\sigma|$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.

Statement 1 is not directly expressible in ZFC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.

But if you have ZFC in the language with a class truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is both expressible and provable in the theory GBC+$T$ is a first-order truth predicate.

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same type. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.

Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any uncountable substructure of $W$, and by Löwenheim-Skolem, let $X$ be any countable elementary substructure of $Y$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.

Statement 1 is not directly expressible in ZFC or GBC, even as a scheme, since you are making explicit reference to the truth of an arbitrary formula $\varphi$. So you need a truth predicate to express the axiom.

But if you have ZFC in the language with a class truth predicate for first-order truth, then I claim statement 1 is provable. That is, it is both expressible and provable in the theory GBC+$T$ is a first-order truth predicate.

To see this, suppose that $W$ is a proper class or even any class of size more than continuum. Since there are only continuum many different types in the language of set theory, it follows that there must be at least two distinct elements $X$ and $Y$ in $W$ with the same type. In other words, $\varphi(X)$ if and only if $\varphi(Y)$ for these two objects.

Unless I have misunderstood, statement 2 seems to be provable in ZFC. If $W$ is a proper class structure with signature $\sigma$. Let $Y$ be any uncountable substructure of $W$, and by Löwenheim-Skolem, let $X$ be any countable elementary substructure of $Y$. So $(X,\sigma)\prec (Y,\sigma)$, as desired.