Theorem rexeqbi1dv 2718

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 2706	. 2 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rexbidv 2509	. 2 |- ( A = B -> ( E. x e. B ph <-> E. x e. B ps ) )
4	1, 3	bitrd 188	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wrex 2487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492

This theorem is referenced by:  reg2exmid  4602  reg3exmid  4646  exmidomni  7270  bj-nn0suc0  16085