Theorem rexeqtrdv 2737

Description: Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)

Hypotheses

Ref	Expression
rexeqtrdv.1	|- ( ph -> E. x e. A ps )
rexeqtrdv.2	|- ( ph -> A = B )

Assertion

Ref	Expression
rexeqtrdv	|- ( ph -> E. x e. B ps )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem rexeqtrdv

Step	Hyp	Ref	Expression
1		rexeqtrdv.1	. 2 |- ( ph -> E. x e. A ps )
2		rexeqtrdv.2	. . 3 |- ( ph -> A = B )
3	2	rexeqdv 2735	. 2 |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) )
4	1, 3	mpbid 147	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   = wceq 1395   E.wrex 2509

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514

This theorem is referenced by: (None)