Theorem rexeqbi1dv 2675

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)

Hypothesis

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

Assertion

Ref	Expression
rexeqbi1dv	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem rexeqbi1dv

Step	Hyp	Ref	Expression
1		rexeq 2667	. 2 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rexbidv 2472	. 2 |- ( A = B -> ( E. x e. B ph <-> E. x e. B ps ) )
4	1, 3	bitrd 187	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1349   E.wrex 2450

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-cleq 2164  df-clel 2167  df-nfc 2302  df-rex 2455

This theorem is referenced by:  reg2exmid  4521  reg3exmid  4565  exmidomni  7122  bj-nn0suc0  14102