Theorem mpgbi 1826

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 1823	. 2 ⊢ ∀𝑥𝜑
3		mpgbi.1	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜓)
4	2, 3	mpbi 233	1 ⊢ 𝜓

Syntax hints:   ↔ wb 209  ∀wal 1566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823

This theorem depends on definitions:  df-bi 210

This theorem is referenced by:  nex  1828  exlimi  2251  axi12  2731  axbnd  2732  nalset  5276  nalsetOLD  5277  bnj1304  35173  bnj1052  35329  bnj1030  35341  bj-nuliota  37637  eu6w  43356  spr0nelg  48170