Theorem rexeq 2750

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 2392	. 2 |- F/_ x A
2		nfcv 2392	. 2 |- F/_ x B
3	1, 2	rexeqf 2746	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534

This theorem is used by:  rexeqi  2754  rexeqdv  2756  rexeqbi1dv  2762  unieq  3944  bnd2  4310  exss  4367  qseq1  6857  finexdc  7207  supeq1  7326  isomni  7476  ismkv  7493  sup3exmid  9287  exmidunben  13317  neifval  15241  cnprcl2k  15307  bj-nn0sucALT  17004  strcoll2  17009  strcollnft  17010  strcollnfALT  17012  sscoll2  17014