Theorem rexeq 2674

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 2319	. 2 |- F/_ x A
2		nfcv 2319	. 2 |- F/_ x B
3	1, 2	rexeqf 2670	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wrex 2456

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461

This theorem is referenced by:  rexeqi  2678  rexeqdv  2680  rexeqbi1dv  2682  unieq  3820  bnd2  4175  exss  4229  qseq1  6585  finexdc  6904  supeq1  6987  isomni  7136  ismkv  7153  sup3exmid  8916  exmidunben  12429  neifval  13725  cnprcl2k  13791  bj-nn0sucALT  14815  strcoll2  14820  strcollnft  14821  strcollnfALT  14823  sscoll2  14825