Consider a theorem of a form $\ A\Rightarrow T.\ $

Consider two proofs:

Proof 1: $$ A\Rightarrow C\qquad\mbox{and} \qquad C\Rightarrow T $$ Proof 2: $$ A\Rightarrow \Gamma\qquad\mbox{and}\qquad \Gamma\Rightarrow T $$

If also another theorem is true:

$$ \neg(C\Rightarrow\Gamma) $$

then -- objectively -- Proof 1 and Proof 2 are not equivalent. And if someone provides a proof of the last-mentioned theorem then Proof 1 and Proof 2 are explicitly not equivalent.

Thus every explicit equivalence is objective but the inverse depends on the status of the last-mentioned theorem.

Consider a theorem of a form $\ A\Rightarrow T.\ $

Consider two proofs:

Proof 1: $$ A\Rightarrow C\qquad\mbox{and} \qquad C\Rightarrow T $$ Proof 2: $$ A\Rightarrow \Gamma\qquad\mbox{and}\qquad \Gamma\Rightarrow T $$

If also another theorem is true:

$$ \neg(C\Rightarrow\Gamma) $$

then -- objectively -- Proof 1 and Proof 2 are not equivalent. And if someone provided proof of the last-mentioned theorem then Proof 1 and Proof 2 would be explicitly not equivalent.

Thus every explicit equivalence is objective but the inverse depends on the status of the last-mentioned theorem.

Consider a theorem of a form $\ A\Rightarrow T.\ $

Consider two proofs:

Proof 1: $$ A\Rightarrow C\qquad\mbox{and} \qquad C\Rightarrow T $$ Proof 2: $$ A\Rightarrow \Gamma\qquad\mbox{and}\qquad \Gamma\Rightarrow T $$

If also another theorem is true:

$$ \neg(C\Rightarrow\Gamma) $$

then -- objectively -- Proof 1 and Proof 2 are not equivalent. And if someone provides a proof of the last-mentioned theorem then Proof 1 and Proof 2 are explicitly not equivalent.

Consider a theorem of a form $\ A\Rightarrow T.\ $

Consider two proofs:

Proof 1: $$ A\Rightarrow C\qquad\mbox{and} \qquad C\Rightarrow T $$ Proof 2: $$ A\Rightarrow \Gamma\qquad\mbox{and}\qquad \Gamma\Rightarrow T $$

If also another theorem is true:

$$ \neg(C\Rightarrow\Gamma) $$

then -- objectively -- Proof 1 and Proof 2 are not equivalent. And if someone provides a proof of the last-mentioned theorem then Proof 1 and Proof 2 are explicitly not equivalent.

Thus every explicit equivalence is objective but the inverse depends on the status of the last-mentioned theorem.