Theorem mprgbir 2433

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 2428	. 2 |- A. x e. A ps
3		mprgbir.1	. 2 |- ( ph <-> A. x e. A ps )
4	2, 3	mpbir 144	1 |- ph

Syntax hints:   -> wi 4   <-> wb 103   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-gen 1383

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

This theorem is referenced by:  ss2rabi  3103  rabnc  3315  ssintub  3706  tron  4209  djussxp  4581  dmiin  4681  dfco2  4930  coiun  4940  tfrlem6  6081  oacl  6221  sbthlem1  6664  peano1nnnn  7387  renfdisj  7544  1nn  8431  ioomax  9364  iccmax  9365  fxnn0nninf  9840  fisumcom2  10828  bezoutlemmain  11261  dfphi2  11470  unennn  11484  znnen  11485  istopon  11565  bj-omtrans2  11807  nninfomnilem  11865  exmidsbthrlem  11867