Theorem cbvex
Rule used to change bound variables, using implicit substitution. Usage
of this theorem is discouraged because it depends on [ax-13]. Check
out cbvexvw, cbvexv1 for weaker versions requiring fewer axioms.
(Contributed by NM, 21-Jun-1993.) (New usage is discouraged.)
Hypotheses
cbval.1
\( \vdash \Finv y \, \varphi \)
cbval.2
\( \vdash \Finv x \, \psi \)
cbval.3
\( \vdash x = y \rightarrow (\varphi \leftrightarrow \psi) \)
Conclusion
\( \vdash \exists x \, \varphi \leftrightarrow \exists y \, \psi \)
Proof
cbval.1
l1
nfn _
cbval.2
l3
nfn _
cbval.3
notbid _
l6
cbval l1 l3 _
\( \vdash \forall x \, \neg \varphi \leftrightarrow \forall y \, \neg \psi \)
l7
alnex
\( \vdash \forall x \, \neg \varphi \leftrightarrow \neg \exists x \, \varphi \)
alnex
\( \vdash \forall y \, \neg \psi \leftrightarrow \neg \exists y \, \psi \)
3bitr3i l6 l7 _
con4bii _