Theorem rexeq 2742

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 2384	. 2 |- F/_ x A
2		nfcv 2384	. 2 |- F/_ x B
3	1, 2	rexeqf 2738	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E.wrex 2521

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526

This theorem is referenced by:  rexeqi  2746  rexeqdv  2748  rexeqbi1dv  2754  unieq  3923  bnd2  4286  exss  4343  qseq1  6817  finexdc  7160  supeq1  7277  isomni  7427  ismkv  7444  sup3exmid  9231  exmidunben  13177  neifval  15005  cnprcl2k  15071  bj-nn0sucALT  16748  strcoll2  16753  strcollnft  16754  strcollnfALT  16756  sscoll2  16758