Theorem rexeq 2703

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 2348	. 2 |- F/_ x A
2		nfcv 2348	. 2 |- F/_ x B
3	1, 2	rexeqf 2699	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wrex 2485

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490

This theorem is referenced by:  rexeqi  2707  rexeqdv  2709  rexeqbi1dv  2715  unieq  3859  bnd2  4217  exss  4271  qseq1  6670  finexdc  6999  supeq1  7088  isomni  7238  ismkv  7255  sup3exmid  9030  exmidunben  12797  neifval  14612  cnprcl2k  14678  bj-nn0sucALT  15914  strcoll2  15919  strcollnft  15920  strcollnfALT  15922  sscoll2  15924