Theorem rexeqbi1dv 2633

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 2625	. 2 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rexbidv 2436	. 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 1331   E.wrex 2415

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420

This theorem is referenced by:  reg2exmid  4446  reg3exmid  4489  exmidomni  7007  bj-nn0suc0  13137