Theorem mprgbir 2685

Description: Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)

Hypotheses

Ref	Expression
mprgbir.1	⊢ (φ ↔ ∀x ∈ A ψ)
mprgbir.2	⊢ (x ∈ A → ψ)

Assertion

Ref	Expression
mprgbir	⊢ φ

Proof of Theorem mprgbir

Step	Hyp	Ref	Expression
1		mprgbir.2	. . 3 ⊢ (x ∈ A → ψ)
2	1	rgen 2680	. 2 ⊢ ∀x ∈ A ψ
3		mprgbir.1	. 2 ⊢ (φ ↔ ∀x ∈ A ψ)
4	2, 3	mpbir 200	1 ⊢ φ

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2615

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  df-ral 2620

This theorem is referenced by:  ss2rabi  3349  rabxm  3574  rabnc  3575  ssintub  3945  pw10  4162  opeq  4620  dmiin  4966  xpnedisj  5514