Theorem rgen2 2619

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 2606	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2586	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   A.wral 2511

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516

This theorem is referenced by:  rgen3  2620  invdisjrab  4087  f1stres  6331  f2ndres  6332  exmidonfinlem  7464  netap  7533  2onetap  7534  2omotaplemap  7536  mpomulf  8229  aptap  8889  divfnzn  9916  fnpfx  11324  wrd2ind  11370  1arith  13020  xpsff1o  13512  mgmidmo  13535  nmznsg  13880  isabli  13967  rhmfn  14267  cnsubmlem  14674  cnsubrglem  14676  txuni2  15067  divcnap  15376  abscncf  15396  recncf  15397  imcncf  15398  cjcncf  15399  reefiso  15588  ioocosf1o  15665  sgmf  15800  perfectlem2  15814  2lgslem1b  15908