Theorem equsalv

Description: An equivalence related to implicit substitution. Version of equsal with a disjoint variable condition, which does not require ax-13 . See equsalvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv . (Contributed by NM, 2-Jun-1993) (Revised by BJ, 31-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	|- F/ x ps
2		equsalv.1	|- ( x = y -> ( ph <-> ps ) )
3	1	19.23	|- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
4	2	pm5.74i	|- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii	|- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		ax6ev	|- E. x x = y
7	6	a1bi	|- ( ps <-> ( E. x x = y -> ps ) )
8	3 5 7	3bitr4i	|- ( A. x ( x = y -> ph ) <-> ps )