Theorem rgen2 2583

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 2570	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2550	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   A.wral 2475

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480

This theorem is referenced by:  rgen3  2584  f1stres  6226  f2ndres  6227  exmidonfinlem  7274  netap  7339  2onetap  7340  2omotaplemap  7342  mpomulf  8035  aptap  8696  divfnzn  9714  1arith  12563  xpsff1o  13053  mgmidmo  13076  nmznsg  13421  isabli  13508  rhmfn  13806  cnsubmlem  14212  cnsubrglem  14214  txuni2  14578  divcnap  14887  abscncf  14907  recncf  14908  imcncf  14909  cjcncf  14910  reefiso  15099  ioocosf1o  15176  sgmf  15308  perfectlem2  15322  2lgslem1b  15416