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  6217  f2ndres  6218  exmidonfinlem  7258  netap  7319  2onetap  7320  2omotaplemap  7322  mpomulf  8014  aptap  8674  divfnzn  9692  1arith  12512  xpsff1o  12968  mgmidmo  12991  nmznsg  13319  isabli  13406  rhmfn  13704  cnsubmlem  14110  cnsubrglem  14112  txuni2  14468  divcnap  14777  abscncf  14797  recncf  14798  imcncf  14799  cjcncf  14800  reefiso  14986  ioocosf1o  15063  sgmf  15194  2lgslem1b  15297