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.