# Metamath Proof Explorer

## Theorem equsex

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsexvw and equsexv for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal . See equsexALT for an alternate proof. (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 6-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 equsex ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)↔{\psi }$

### 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 2 biimpa ${⊢}\left({x}={y}\wedge {\phi }\right)\to {\psi }$
4 1 3 exlimi ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)\to {\psi }$
5 1 2 equsal ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)↔{\psi }$
6 equs4 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\to \exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)$
7 5 6 sylbir ${⊢}{\psi }\to \exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)$
8 4 7 impbii ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)↔{\psi }$