Theorem cbval2v

Rule used to change bound variables, using implicit substitution. Version of cbval2 with a disjoint variable condition, which does not require [ax-13]. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by GG, 10-Jan-2024.)

Hypotheses
cbval2v.1
\( \vdash \Finv z \, \varphi \)
cbval2v.2
\( \vdash \Finv w \, \varphi \)
cbval2v.3
\( \vdash \Finv x \, \psi \)
cbval2v.4
\( \vdash \Finv y \, \psi \)
cbval2v.5
\( \vdash x = z \wedge y = w \rightarrow (\varphi \leftrightarrow \psi) \)
Conclusion
\( \vdash \forall x \, \forall y \, \varphi \leftrightarrow \forall z \, \forall w \, \psi \)
Proof
cbval2v.1
l1
nfal _
cbval2v.3
l3
nfal _
l4
nfv
l5
nfv
cbval2v.2
l7
a1i _
cbval2v.4
l9
a1i _
cbval2v.5
ex _
cbv2w l4 l5 l7 l9 _
cbvalv1 l1 l3 _