Theorem mprg 2489

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 2485	. 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 1480   A.wral 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425

This theorem depends on definitions:  df-bi 116  df-ral 2421

This theorem is referenced by:  reximia  2527  rmoimia  2886  iuneq2i  3831  iineq2i  3832  dfiun2  3847  dfiin2  3848  dfiun3  4798  dfiin3  4799  cnviinm  5080  ixpintm  6619  sumeq2i  11136  prodeq2i  11334  bj-omtrans  13157