Theorem bj-cbvexw 32664

Description: Change bound variable. This is to cbvexvw 1970 what cbvalw 1968 is to cbvalvw 1969. (Contributed by BJ, 17-Mar-2020.)

Hypotheses

Ref	Expression
bj-cbvexw.1	|- ( E. x E. y ps -> E. y ps )
bj-cbvexw.2	|- ( ph -> A. y ph )
bj-cbvexw.3	|- ( E. y E. x ph -> E. x ph )
bj-cbvexw.4	|- ( ps -> A. x ps )
bj-cbvexw.5	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem bj-cbvexw

Step	Hyp	Ref	Expression
1		bj-cbvexw.1	. . 3 |- ( E. x E. y ps -> E. y ps )
2		bj-cbvexw.2	. . 3 |- ( ph -> A. y ph )
3		bj-cbvexw.5	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
4	3	equcoms 1947	. . . 4 |- ( y = x -> ( ph <-> ps ) )
5	4	biimpd 219	. . 3 |- ( y = x -> ( ph -> ps ) )
6	1, 2, 5	bj-cbvexiw 32659	. 2 |- ( E. x ph -> E. y ps )
7		bj-cbvexw.3	. . 3 |- ( E. y E. x ph -> E. x ph )
8		bj-cbvexw.4	. . 3 |- ( ps -> A. x ps )
9	3	biimprd 238	. . 3 |- ( x = y -> ( ps -> ph ) )
10	7, 8, 9	bj-cbvexiw 32659	. 2 |- ( E. y ps -> E. x ph )
11	6, 10	impbii 199	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)