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 x ψ
equsal.2 x = y φ ψ
Assertion equsex x x = y φ ψ


Step Hyp Ref Expression
1 equsal.1 x ψ
2 equsal.2 x = y φ ψ
3 2 biimpa x = y φ ψ
4 1 3 exlimi x x = y φ ψ
5 1 2 equsal x x = y φ ψ
6 equs4 x x = y φ x x = y φ
7 5 6 sylbir ψ x x = y φ
8 4 7 impbii x x = y φ ψ