Theorem rgen2 2592

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 2579	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2559	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   A.wral 2484

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489

This theorem is referenced by:  rgen3  2593  invdisjrab  4039  f1stres  6247  f2ndres  6248  exmidonfinlem  7303  netap  7368  2onetap  7369  2omotaplemap  7371  mpomulf  8064  aptap  8725  divfnzn  9744  1arith  12723  xpsff1o  13214  mgmidmo  13237  nmznsg  13582  isabli  13669  rhmfn  13967  cnsubmlem  14373  cnsubrglem  14375  txuni2  14761  divcnap  15070  abscncf  15090  recncf  15091  imcncf  15092  cjcncf  15093  reefiso  15282  ioocosf1o  15359  sgmf  15491  perfectlem2  15505  2lgslem1b  15599