Theorem rexeqtrrdv 2741

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

Hypotheses

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

Assertion

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

Distinct variable groups:   x, A   x, B

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

Proof of Theorem rexeqtrrdv

Step	Hyp	Ref	Expression
1		rexeqtrrdv.1	. 2 |- ( ph -> E. x e. A ps )
2		rexeqtrrdv.2	. . 3 |- ( ph -> B = A )
3	2	rexeqdv 2737	. 2 |- ( ph -> ( E. x e. B ps <-> E. x e. A ps ) )
4	1, 3	mpbird 167	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   = wceq 1397   E.wrex 2511

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516

This theorem is referenced by: (None)