# Metamath Proof Explorer

## Theorem equsal

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsalvw and equsalv for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex . (Contributed by NM, 2-Jun-1993) (Proof shortened by Andrew Salmon, 12-Aug-2011) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 5-Feb-2018) (New usage is discouraged.)

Ref Expression
Hypotheses equsal.1 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
equsal.2 ${⊢}{x}={y}\to \left({\phi }↔{\psi }\right)$
Assertion equsal ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)↔{\psi }$

### Proof

Step Hyp Ref Expression
1 equsal.1 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\psi }$
2 equsal.2 ${⊢}{x}={y}\to \left({\phi }↔{\psi }\right)$
3 1 19.23 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\psi }\right)↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to {\psi }\right)$
4 2 pm5.74i ${⊢}\left({x}={y}\to {\phi }\right)↔\left({x}={y}\to {\psi }\right)$
5 4 albii ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\psi }\right)$
6 ax6e ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}{x}={y}$
7 6 a1bi ${⊢}{\psi }↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to {\psi }\right)$
8 3 5 7 3bitr4i ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)↔{\psi }$