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Metamath Proof Explorer

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