Theorem rgen2w 2586

Description: Generalization rule for restricted quantification. Note that x and y needn't be distinct. (Contributed by NM, 18-Jun-2014.)

Hypothesis

Ref	Expression
rgenw.1	|- ph

Assertion

Ref	Expression
rgen2w	|- A. x e. A A. y e. B ph

Proof of Theorem rgen2w

Step	Hyp	Ref	Expression
1		rgenw.1	. . 3 |- ph
2	1	rgenw 2585	. 2 |- A. y e. B ph
3	2	rgenw 2585	1 |- A. x e. A A. y e. B ph

Syntax hints:   A.wral 2508

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-gen 1495

This theorem depends on definitions:  df-bi 117  df-ral 2513

This theorem is referenced by:  fnmpoi  6349  ixxf  10094  fzf  10208  rexfiuz  11500  prdsvallem  13305  eltx  14933  txcnp  14945  txcnmpt  14947  txrest  14950  txlm  14953