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    )