Theorem rgen2 2459

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 2446	. 2 |- ( x e. A -> A. y e. B ph )
3	2	rgen 2428	1 |- A. x e. A A. y e. B ph

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1438   A.wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364

This theorem is referenced by:  rgen3  2460  f1stres  5922  f2ndres  5923  divfnzn  9096  abscncf  11524  recncf  11525  imcncf  11526  cjcncf  11527