Theorem cbvexvw 1969

Description: Change bound variable. See cbvexv 1967 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1496. (Revised by GG, 25-Aug-2024.)

Hypothesis

Ref	Expression
cbvalvw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, y   ps, x   ph, y

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

Proof of Theorem cbvexvw

Step	Hyp	Ref	Expression
1		cbvalvw.1	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
2	1	biimpd 144	. . . . 5 |- ( x = y -> ( ph -> ps ) )
3	2	equcoms 1756	. . . 4 |- ( y = x -> ( ph -> ps ) )
4	3	spimev 1909	. . 3 |- ( ph -> E. y ps )
5	4	exlimiv 1646	. 2 |- ( E. x ph -> E. y ps )
6	1	biimprd 158	. . . 4 |- ( x = y -> ( ps -> ph ) )
7	6	spimev 1909	. . 3 |- ( ps -> E. x ph )
8	7	exlimiv 1646	. 2 |- ( E. y ps -> E. x ph )
9	5, 8	impbii 126	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1540

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509

This theorem is referenced by:  cbvex2vw  1982  cbvrexvw  2772  prodmodc  12138