Theorem rexeqd 1792

Description: Equality deduction for restricted existential quantifier.

Hypothesis

Ref	Expression
raleqd.1	|- (A = B -> (ph <-> ps))

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem rexeqd

Step	Hyp	Ref	Expression
1		rexeq1 1787	. 2 |- (A = B -> (E.x e. A ph <-> E.x e. B ph))
2		raleqd.1	. . 3 |- (A = B -> (ph <-> ps))
3	2	rexbidv 1664	. 2 |- (A = B -> (E.x e. B ph <-> E.x e. B ps))
4	1, 3	bitrd 528	1 |- (A = B -> (E.x e. A ph <-> E.x e. B ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956  E.wrex 1646

This theorem is referenced by:  fri 2918  frc 2920  isofrlem 3901  f1oweALT 3906  zfregcl 4595  ishaus 7783  isgrp 8041  spwval 8659  spwnex 8661  pjtht 9234

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-rex 1650

Copyright terms: Public domain