Theorem mprg 2562

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 2558	. 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 2175   A.wral 2483

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 1471

This theorem depends on definitions:  df-bi 117  df-ral 2488

This theorem is referenced by:  reximia  2600  rmoimia  2974  iuneq2i  3944  iineq2i  3945  dfiun2  3960  dfiin2  3961  dfiun3  4935  dfiin3  4936  cnviinm  5221  ixpintm  6802  sumeq2i  11594  prodeq2i  11792  2sqlem1  15509  bj-omtrans  15756