Theorem rexeq 2671

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 2317	. 2 |- F/_ x A
2		nfcv 2317	. 2 |- F/_ x B
3	1, 2	rexeqf 2667	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wrex 2454

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459

This theorem is referenced by:  rexeqi  2675  rexeqdv  2677  rexeqbi1dv  2679  unieq  3814  bnd2  4168  exss  4221  qseq1  6573  finexdc  6892  supeq1  6975  isomni  7124  ismkv  7141  sup3exmid  8887  exmidunben  12394  neifval  13211  cnprcl2k  13277  bj-nn0sucALT  14290  strcoll2  14295  strcollnft  14296  strcollnfALT  14298  sscoll2  14300