Theorem rgen2 2628

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 2615	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2595	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   A.wral 2520

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 2525

This theorem is referenced by:  rgen3  2629  invdisjrab  4103  f1stres  6353  f2ndres  6354  exmidonfinlem  7496  netap  7568  2onetap  7569  2omotaplemap  7571  mpomulf  8264  aptap  8924  divfnzn  9953  fnpfx  11369  wrd2ind  11415  1arith  13065  ballotfilem2  13142  xpsff1o  13562  mgmidmo  13585  nmznsg  13930  isabli  14017  rhmfn  14317  cnsubmlem  14726  cnsubrglem  14728  txuni2  15121  divcnap  15430  abscncf  15450  recncf  15451  imcncf  15452  cjcncf  15453  reefiso  15642  ioocosf1o  15719  sgmf  15854  perfectlem2  15868  2lgslem1b  15962