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	|- F/ x ps
	equsal.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	equsex	|- ( E. x ( x = y /\ ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		equsal.1	|- F/ x ps
2		equsal.2	|- ( x = y -> ( ph <-> ps ) )
3	2	biimpa	|- ( ( x = y /\ ph ) -> ps )
4	1 3	exlimi	|- ( E. x ( x = y /\ ph ) -> ps )
5	1 2	equsal	|- ( A. x ( x = y -> ph ) <-> ps )
6		equs4	|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
7	5 6	sylbir	|- ( ps -> E. x ( x = y /\ ph ) )
8	4 7	impbii	|- ( E. x ( x = y /\ ph ) <-> ps )