Theorem mprgbir 2588

Description: Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)

Hypotheses

Ref	Expression
mprgbir.1	|- ( ph <-> A. x e. A ps )
mprgbir.2	|- ( x e. A -> ps )

Assertion

Ref	Expression
mprgbir	|- ph

Proof of Theorem mprgbir

Step	Hyp	Ref	Expression
1		mprgbir.2	. . 3 |- ( x e. A -> ps )
2	1	rgen 2583	. 2 |- A. x e. A ps
3		mprgbir.1	. 2 |- ( ph <-> A. x e. A ps )
4	2, 3	mpbir 146	1 |- ph

Syntax hints:   -> wi 4   <-> wb 105   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:  ss2rabi  3306  rabnc  3524  ssintub  3941  tron  4473  djussxp  4867  dmiin  4970  dfco2  5228  coiun  5238  tfrlem6  6468  oacl  6614  sbthlem1  7135  peano1nnnn  8050  renfdisj  8217  1nn  9132  ioomax  10156  iccmax  10157  xnn0nnen  10671  fxnn0nninf  10673  fisumcom2  11965  fprodcom2fi  12153  bezoutlemmain  12535  dfphi2  12758  unennn  12984  znnen  12985  istopon  14703  neipsm  14844  lgsquadlem2  15773  pw0ss  15899  clwwlkn0  16151  bj-omtrans2  16403  nninfomnilem  16472  exmidsbthrlem  16478