Theorem rexeq 2732

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 2375	. 2 |- F/_ x A
2		nfcv 2375	. 2 |- F/_ x B
3	1, 2	rexeqf 2728	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E.wrex 2512

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517

This theorem is referenced by:  rexeqi  2736  rexeqdv  2738  rexeqbi1dv  2744  unieq  3907  bnd2  4269  exss  4325  qseq1  6795  finexdc  7135  supeq1  7245  isomni  7395  ismkv  7412  sup3exmid  9196  exmidunben  13127  neifval  14951  cnprcl2k  15017  bj-nn0sucALT  16694  strcoll2  16699  strcollnft  16700  strcollnfALT  16702  sscoll2  16704