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Metamath Proof Explorer

Theorem mpgbi

Description: Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994) (Proof shortened by Stefan Allan, 28-Oct-2008)

	Ref	Expression
Hypotheses	mpgbi.1	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜓 )
	mpgbi.2	⊢ 𝜑
Assertion	mpgbi	⊢ 𝜓

Proof

Step	Hyp	Ref	Expression
1		mpgbi.1	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜓 )
2		mpgbi.2	⊢ 𝜑
3	2	ax-gen	⊢ ∀ 𝑥 𝜑
4	3 1	mpbi	⊢ 𝜓