Theorem rexeq 2706

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem rexeq

Step	Hyp	Ref	Expression
1		nfcv 2350	. 2 |- F/_ x A
2		nfcv 2350	. 2 |- F/_ x B
3	1, 2	rexeqf 2702	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

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:  rexeqi  2710  rexeqdv  2712  rexeqbi1dv  2718  unieq  3873  bnd2  4233  exss  4289  qseq1  6693  finexdc  7025  supeq1  7114  isomni  7264  ismkv  7281  sup3exmid  9065  exmidunben  12912  neifval  14727  cnprcl2k  14793  bj-nn0sucALT  16113  strcoll2  16118  strcollnft  16119  strcollnfALT  16121  sscoll2  16123