Theorem mpgbi 1549

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	⊢ (∀xφ ↔ ψ)
mpgbi.2	⊢ φ

Assertion

Ref	Expression
mpgbi	⊢ ψ

Proof of Theorem mpgbi

Step	Hyp	Ref	Expression
1		mpgbi.2	. . 3 ⊢ φ
2	1	ax-gen 1546	. 2 ⊢ ∀xφ
3		mpgbi.1	. 2 ⊢ (∀xφ ↔ ψ)
4	2, 3	mpbi 199	1 ⊢ ψ

Syntax hints:   ↔ wb 176  ∀wal 1540

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

This theorem is referenced by:  nex  1555  exlimi  1803  exlimihOLD  1805  exan  1882  abbii  2466