Theorem cbvexv1 1766

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex 1770 with a disjoint variable condition. See cbvexvw 1935 for a version with two disjoint variable conditions, and cbvexv 1933 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	|- F/ y ph
cbvalv1.nf2	|- F/ x ps
cbvalv1.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvexv1	|- ( E. x ph <-> E. y ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvexv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf2	. . . 4 |- F/ x ps
2	1	nfex 1651	. . 3 |- F/ x E. y ps
3		cbvalv1.nf1	. . . . . 6 |- F/ y ph
4	3	nfri 1533	. . . . 5 |- ( ph -> A. y ph )
5		cbvalv1.1	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
6	5	bicomd 141	. . . . . 6 |- ( x = y -> ( ps <-> ph ) )
7	6	equcoms 1722	. . . . 5 |- ( y = x -> ( ps <-> ph ) )
8	4, 7	equsex 1742	. . . 4 |- ( E. y ( y = x /\ ps ) <-> ph )
9		exsimpr 1632	. . . 4 |- ( E. y ( y = x /\ ps ) -> E. y ps )
10	8, 9	sylbir 135	. . 3 |- ( ph -> E. y ps )
11	2, 10	exlimi 1608	. 2 |- ( E. x ph -> E. y ps )
12	3	nfex 1651	. . 3 |- F/ y E. x ph
13	1	nfri 1533	. . . . 5 |- ( ps -> A. x ps )
14	13, 5	equsex 1742	. . . 4 |- ( E. x ( x = y /\ ph ) <-> ps )
15		exsimpr 1632	. . . 4 |- ( E. x ( x = y /\ ph ) -> E. x ph )
16	14, 15	sylbir 135	. . 3 |- ( ps -> E. x ph )
17	12, 16	exlimi 1608	. 2 |- ( E. y ps -> E. x ph )
18	11, 17	impbii 126	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1474   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  cbvrexfw  2720  fprod2dlemstep  11787