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  6218  f2ndres  6219  exmidonfinlem  7262  netap  7323  2onetap  7324  2omotaplemap  7326  mpomulf  8018  aptap  8679  divfnzn  9697  1arith  12546  xpsff1o  13002  mgmidmo  13025  nmznsg  13353  isabli  13440  rhmfn  13738  cnsubmlem  14144  cnsubrglem  14146  txuni2  14502  divcnap  14811  abscncf  14831  recncf  14832  imcncf  14833  cjcncf  14834  reefiso  15023  ioocosf1o  15100  sgmf  15232  perfectlem2  15246  2lgslem1b  15340