# Metamath Proof Explorer

## Theorem cbvalvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Apr-2017) (Proof shortened by Wolf Lammen, 28-Feb-2018)

Ref Expression
Hypothesis cbvalvw.1 ${⊢}{x}={y}\to \left({\phi }↔{\psi }\right)$
Assertion cbvalvw ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }↔\forall {y}\phantom{\rule{.4em}{0ex}}{\psi }$

### Proof

Step Hyp Ref Expression
1 cbvalvw.1 ${⊢}{x}={y}\to \left({\phi }↔{\psi }\right)$
2 ax-5 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }$
3 ax-5 ${⊢}¬{\psi }\to \forall {x}\phantom{\rule{.4em}{0ex}}¬{\psi }$
4 ax-5 ${⊢}\forall {y}\phantom{\rule{.4em}{0ex}}{\psi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\forall {y}\phantom{\rule{.4em}{0ex}}{\psi }$
5 ax-5 ${⊢}¬{\phi }\to \forall {y}\phantom{\rule{.4em}{0ex}}¬{\phi }$
6 2 3 4 5 1 cbvalw ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{\phi }↔\forall {y}\phantom{\rule{.4em}{0ex}}{\psi }$