Theorem rexeq 2630

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 2282	. 2 |- F/_ x A
2		nfcv 2282	. 2 |- F/_ x B
3	1, 2	rexeqf 2626	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   E.wrex 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423

This theorem is referenced by:  rexeqi  2634  rexeqdv  2636  rexeqbi1dv  2638  unieq  3753  bnd2  4105  exss  4157  qseq1  6485  finexdc  6804  supeq1  6881  isomni  7016  ismkv  7035  sup3exmid  8739  exmidunben  11975  neifval  12348  cnprcl2k  12414  bj-nn0sucALT  13347  strcoll2  13352  strcollnft  13353  strcollnfALT  13355  sscoll2  13357