Theorem mpgbi 1873

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 1870	. 2 ⊢ ∀𝑥𝜑
3		mpgbi.1	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜓)
4	2, 3	mpbi 220	1 ⊢ 𝜓

Syntax hints:   ↔ wb 196  ∀wal 1629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  nex  1879  exlimi  2242  exlimiOLD  2383  axi12  2749  abbii  2888  nalset  4929  bnj1304  31228  bnj1052  31381  bnj1030  31393  bj-abbii  33113  bj-nalset  33130  bj-nuliota  33350  spr0nelg  42254