Theorem cbvexdw
Deduction used to change bound variables, using implicit substitution.
Version of cbvexd with a disjoint variable condition, which does not
require [ax-13]. (Contributed by NM, 2-Jan-2002.) Avoid [ax-13].
(Revised by GG, 10-Jan-2024.)
Hypotheses
cbvaldw.1
\( \vdash \Finv y \, \varphi \)
cbvaldw.2
\( \vdash \varphi \rightarrow \Finv y \, \psi \)
cbvaldw.3
\( \vdash \varphi \rightarrow x = y \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\exists x \, \psi \leftrightarrow \exists y \, \chi) \)
Proof
l0
cbvaldw.1
cbvaldw.2
l2
nfnd _
l3
cbvaldw.3
\( \vdash \varphi \rightarrow x = y \rightarrow (\psi \leftrightarrow \chi) \)
notbi
\( \vdash \psi \leftrightarrow \chi \leftrightarrow (\neg \psi \leftrightarrow \neg \chi) \)
imbitrdi l3 _
l6
cbvaldw l0 l2 _
\( \vdash \varphi \rightarrow (\forall x \, \neg \psi \leftrightarrow \forall y \, \neg \chi) \)
l7
alnex
\( \vdash \forall x \, \neg \psi \leftrightarrow \neg \exists x \, \psi \)
alnex
\( \vdash \forall y \, \neg \chi \leftrightarrow \neg \exists y \, \chi \)
3bitr3g l6 l7 _
con4bid _