Theorem equsexv

Description: An equivalence related to implicit substitution. Version of equsex with a disjoint variable condition, which does not require ax-13 . See equsexvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	equsalv.nf	|- F/ x ps
	equsalv.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	equsexv	|- ( E. x ( x = y /\ ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	|- F/ x ps
2		equsalv.1	|- ( x = y -> ( ph <-> ps ) )
3		sbalex	|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
4	1 2	equsalv	|- ( A. x ( x = y -> ph ) <-> ps )
5	3 4	bitri	|- ( E. x ( x = y /\ ph ) <-> ps )