Theorem bj-cbvexiw 32659

Description: Change bound variable. This is to cbvexvw 1970 what cbvaliw 1933 is to cbvalvw 1969. [TODO: move after cbvalivw 1934]. (Contributed by BJ, 17-Mar-2020.)

Hypotheses

Ref	Expression
bj-cbvexiw.1	|- ( E. x E. y ps -> E. y ps )
bj-cbvexiw.2	|- ( ph -> A. y ph )
bj-cbvexiw.3	|- ( y = x -> ( ph -> ps ) )

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem bj-cbvexiw

Step	Hyp	Ref	Expression
1		bj-cbvexiw.1	. 2 |- ( E. x E. y ps -> E. y ps )
2		bj-cbvexiw.2	. . 3 |- ( ph -> A. y ph )
3		bj-cbvexiw.3	. . 3 |- ( y = x -> ( ph -> ps ) )
4	2, 3	spimeh 1925	. 2 |- ( ph -> E. y ps )
5	1, 4	bj-exlime 32609	1 |- ( E. x ph -> E. y ps )

Syntax hints:   -> wi 4   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-6 1888

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

This theorem is referenced by:  bj-cbvexivw  32660  bj-cbvexw  32664