Theorem rgen2 2618

Description: Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)

Hypothesis

Ref	Expression
rgen2.1	|- ( ( x e. A /\ y e. B ) -> ph )

Assertion

Ref	Expression
rgen2	|- A. x e. A A. y e. B ph

Distinct variable groups:   x, y   y, A

Allowed substitution hints:   ph( x, y)   A( x)   B( x, y)

Proof of Theorem rgen2

Step	Hyp	Ref	Expression
1		rgen2.1	. . 3 |- ( ( x e. A /\ y e. B ) -> ph )
2	1	ralrimiva 2605	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2585	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   A.wral 2510

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-5 1495  ax-gen 1497  ax-4 1558  ax-17 1574

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515

This theorem is referenced by:  rgen3  2619  invdisjrab  4082  f1stres  6322  f2ndres  6323  exmidonfinlem  7404  netap  7473  2onetap  7474  2omotaplemap  7476  mpomulf  8169  aptap  8830  divfnzn  9855  fnpfx  11262  wrd2ind  11308  1arith  12958  xpsff1o  13450  mgmidmo  13473  nmznsg  13818  isabli  13905  rhmfn  14205  cnsubmlem  14611  cnsubrglem  14613  txuni2  14999  divcnap  15308  abscncf  15328  recncf  15329  imcncf  15330  cjcncf  15331  reefiso  15520  ioocosf1o  15597  sgmf  15729  perfectlem2  15743  2lgslem1b  15837