Theorem rexeq 2731

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 2374	. 2 |- F/_ x A
2		nfcv 2374	. 2 |- F/_ x B
3	1, 2	rexeqf 2727	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   E.wrex 2511

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516

This theorem is referenced by:  rexeqi  2735  rexeqdv  2737  rexeqbi1dv  2743  unieq  3902  bnd2  4263  exss  4319  qseq1  6751  finexdc  7091  supeq1  7184  isomni  7334  ismkv  7351  sup3exmid  9136  exmidunben  13046  neifval  14863  cnprcl2k  14929  bj-nn0sucALT  16573  strcoll2  16578  strcollnft  16579  strcollnfALT  16581  sscoll2  16583