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Metamath Proof Explorer

Theorem gen11

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv is gen11 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis gen11.1 ⊢ (    𝜑    ▶    𝜓    )
Assertion gen11 (    𝜑    ▶    ∀ 𝑥 𝜓    )

Proof

Step Hyp Ref Expression
1 gen11.1 ⊢ (    𝜑    ▶    𝜓    )
2 dfvd1imp ⊢ ( (    𝜑    ▶    𝜓    ) → ( 𝜑 → 𝜓 ) )
3 1 2 ax-mp ⊢ ( 𝜑 → 𝜓 )
4 3 alrimiv ⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
5 dfvd1impr ⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) → (    𝜑    ▶    ∀ 𝑥 𝜓    ) )
6 4 5 ax-mp ⊢ (    𝜑    ▶    ∀ 𝑥 𝜓    )