Theorem mprg 2544

Description: Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)

Hypotheses

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

Assertion

Ref	Expression
mprg	|- ps

Proof of Theorem mprg

Step	Hyp	Ref	Expression
1		mprg.2	. . 3 |- ( x e. A -> ph )
2	1	rgen 2540	. 2 |- A. x e. A ph
3		mprg.1	. 2 |- ( A. x e. A ph -> ps )
4	2, 3	ax-mp 5	1 |- ps

Syntax hints:   -> wi 4   e. wcel 2158   A.wral 2465

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 1459

This theorem depends on definitions:  df-bi 117  df-ral 2470

This theorem is referenced by:  reximia  2582  rmoimia  2951  iuneq2i  3916  iineq2i  3917  dfiun2  3932  dfiin2  3933  dfiun3  4898  dfiin3  4899  cnviinm  5182  ixpintm  6739  sumeq2i  11386  prodeq2i  11584  2sqlem1  14757  bj-omtrans  15004