Theorem rexeqbidv 2678

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)

Hypotheses

Ref	Expression
raleqbidv.1	|- ( ph -> A = B )
raleqbidv.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
rexeqbidv	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Distinct variable groups:   x, A   x, B   ph, x

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

Proof of Theorem rexeqbidv

Step	Hyp	Ref	Expression
1		raleqbidv.1	. . 3 |- ( ph -> A = B )
2	1	rexeqdv 2672	. 2 |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) )
3		raleqbidv.2	. . 3 |- ( ph -> ( ps <-> ch ) )
4	3	rexbidv 2471	. 2 |- ( ph -> ( E. x e. B ps <-> E. x e. B ch ) )
5	2, 4	bitrd 187	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  supeq123d  6968  ismnddef  12654  mndpropd  12676  mnd1  12679  isgrp  12714  isgrpd2e  12726