Theorem mpgbi 1725

Description: Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)

Hypotheses

Ref	Expression
mpgbi.1	|- ( A. x ph <-> ps )
mpgbi.2	|- ph

Assertion

Ref	Expression
mpgbi	|- ps

Proof of Theorem mpgbi

Step	Hyp	Ref	Expression
1		mpgbi.2	. . 3 |- ph
2	1	ax-gen 1722	. 2 |- A. x ph
3		mpgbi.1	. 2 |- ( A. x ph <-> ps )
4	2, 3	mpbi 220	1 |- ps

Syntax hints:   <-> wb 196   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  nex  1731  exlimi  2086  exlimiOLD  2221  axi12  2600  abbii  2739  nalset  4795  bnj1304  30890  bnj1052  31043  bnj1030  31055  bj-abbii  32777  bj-nalset  32794  bj-nuliota  33019  spr0nelg  41726