Theorem mpgbi 1381

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 1378	. 2 ⊢ ∀𝑥𝜑
3		mpgbi.1	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜓)
4	2, 3	mpbi 143	1 ⊢ 𝜓

Syntax hints:   ↔ wb 103  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-gen 1378

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nex  1429  exlimih  1524  exan  1623  abbii  2194  bj-ex  10573