Theorem mprg 2590

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 2586	. 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 2202   A.wral 2511

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 1498

This theorem depends on definitions:  df-bi 117  df-ral 2516

This theorem is referenced by:  reximia  2628  rmoimia  3009  iuneq2i  3993  iineq2i  3994  dfiun2  4009  dfiin2  4010  dfiun3  4997  dfiin3  4998  cnviinm  5285  ixpintm  6937  sumeq2i  11987  prodeq2i  12186  2sqlem1  15916  bj-omtrans  16655