Theorem mprg 2514

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 2510	. 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 2128   A.wral 2435

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 1429

This theorem depends on definitions:  df-bi 116  df-ral 2440

This theorem is referenced by:  reximia  2552  rmoimia  2914  iuneq2i  3869  iineq2i  3870  dfiun2  3885  dfiin2  3886  dfiun3  4847  dfiin3  4848  cnviinm  5129  ixpintm  6672  sumeq2i  11272  prodeq2i  11470  bj-omtrans  13602