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Metamath Proof Explorer

Theorem mpgbir

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	mpgbir.1	⊢ ( 𝜑 ↔ ∀ 𝑥 𝜓 )
	mpgbir.2	⊢ 𝜓
Assertion	mpgbir	⊢ 𝜑

Proof

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