Theorem rexeqd 1328

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 group(s):   x,A   x,B

Proof of Theorem rexeqd

Step	Hyp	Ref	Expression
1		rexeq 1325	. 2 |- (A = B -> (E.x e. A ph <-> E.x e. B ph))
2		raleqd.1	. . 3 |- (A = B -> (ph <-> ps))
3	2	birexdv 1220	. 2 |- (A = B -> (E.x e. B ph <-> E.x e. B ps))
4	1, 3	bitrd 406	1 |- (A = B -> (E.x e. A ph <-> E.x e. B ps))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091  E.wrex 1202

This theorem is referenced by:  fri 2170  frc 2172  isofrlem 2939  f1oweOLD 2944  zfregcl 3446  pjtht 5240

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-rex 1206

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