Theorem mprg 2684

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 2680	. 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 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  reximia  2720  rmoimia  3037  iuneq2i  3988  iineq2i  3989  dfiun2  4002  dfiin2  4003  fnpw1fn  5854  dfnnc3  5886  enmap2lem2  6065  enmap1lem2  6071  enprmaplem2  6078  enprmaplem5  6081  fntcfn  6246  fnspac  6284  fnfreclem3  6320