Theorem mprgbir

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

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

Proof

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