Theorem rexeq 2750

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 2392	. 2 |- F/_ x A
2		nfcv 2392	. 2 |- F/_ x B
3	1, 2	rexeqf 2746	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

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:  rexeqi  2754  rexeqdv  2756  rexeqbi1dv  2762  unieq  3939  bnd2  4305  exss  4362  qseq1  6847  finexdc  7197  supeq1  7316  isomni  7466  ismkv  7483  sup3exmid  9277  exmidunben  13295  neifval  15164  cnprcl2k  15230  bj-nn0sucALT  16918  strcoll2  16923  strcollnft  16924  strcollnfALT  16926  sscoll2  16928