Theorem mprg 2564

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 2560	. 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 2177   A.wral 2485

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 1473

This theorem depends on definitions:  df-bi 117  df-ral 2490

This theorem is referenced by:  reximia  2602  rmoimia  2979  iuneq2i  3951  iineq2i  3952  dfiun2  3967  dfiin2  3968  dfiun3  4946  dfiin3  4947  cnviinm  5233  ixpintm  6825  sumeq2i  11750  prodeq2i  11948  2sqlem1  15666  bj-omtrans  16030