Theorem rexeqbi1dv 2762

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 2750	. 2 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rexbidv 2551	. 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 1402   E.wrex 2529

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534

This theorem is referenced by:  reg2exmid  4678  reg3exmid  4722  exmidomni  7472  bj-nn0suc0  16890