Theorem rgenw 2585

Description: Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)

Hypothesis

Ref	Expression
rgenw.1	|- ph

Assertion

Ref	Expression
rgenw	|- A. x e. A ph

Proof of Theorem rgenw

Step	Hyp	Ref	Expression
1		rgenw.1	. . 3 |- ph
2	1	a1i 9	. 2 |- ( x e. A -> ph )
3	2	rgen 2583	1 |- A. x e. A ph

Syntax hints:   e. wcel 2200   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:  rgen2w  2586  reuun1  3486  0disj  4080  iinexgm  4238  epse  4433  xpiindim  4859  eliunxp  4861  opeliunxp2  4862  elrnmpti  4977  fnmpti  5452  mpoeq12  6070  relmptopab  6213  iunex  6274  mpoex  6366  opeliunxp2f  6390  ixpssmap  6887  1domsn  6984  nneneq  7026  nqprrnd  7738  nqprdisj  7739  uzf  9733  sum0  11907  fisumcom2  11957  prod0  12104  fprodcom2fi  12145  phisum  12771  sumhashdc  12878  unennn  12976  prdsvallem  13313  prdsval  13314  fczpsrbag  14643  psr1clfi  14660  tgidm  14756  tgrest  14851  txbas  14940  reldvg  15361  dvfvalap  15363  bj-indint  16318  bj-nn0suc0  16337  bj-nntrans  16338