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	2	pm5.32i	|- ( ( x = y /\ ph ) <-> ( x = y /\ ps ) )
4	3	exbii	|- ( E. x ( x = y /\ ph ) <-> E. x ( x = y /\ ps ) )
5		ax6ev	|- E. x x = y
6	1	19.41	|- ( E. x ( x = y /\ ps ) <-> ( E. x x = y /\ ps ) )
7	5 6	mpbiran	|- ( E. x ( x = y /\ ps ) <-> ps )
8	4 7	bitri	|- ( E. x ( x = y /\ ph ) <-> ps )