Theorem rgen2 2552

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 2539	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2519	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449

This theorem is referenced by:  rgen3  2553  f1stres  6127  f2ndres  6128  exmidonfinlem  7149  divfnzn  9559  1arith  12297  mgmidmo  12603  txuni2  12896  divcnap  13195  abscncf  13212  recncf  13213  imcncf  13214  cjcncf  13215  reefiso  13338  ioocosf1o  13415