Theorem rgen2 2594

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 2581	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2561	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   A.wral 2486

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2491

This theorem is referenced by:  rgen3  2595  invdisjrab  4053  f1stres  6268  f2ndres  6269  exmidonfinlem  7332  netap  7401  2onetap  7402  2omotaplemap  7404  mpomulf  8097  aptap  8758  divfnzn  9777  fnpfx  11168  wrd2ind  11214  1arith  12805  xpsff1o  13296  mgmidmo  13319  nmznsg  13664  isabli  13751  rhmfn  14049  cnsubmlem  14455  cnsubrglem  14457  txuni2  14843  divcnap  15152  abscncf  15172  recncf  15173  imcncf  15174  cjcncf  15175  reefiso  15364  ioocosf1o  15441  sgmf  15573  perfectlem2  15587  2lgslem1b  15681