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Metamath Proof Explorer

Theorem gen11nv

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih is gen11nv without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	gen11nv.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	gen11nv.2	⊢ ( 𝜑 ▶ 𝜓 )
Assertion	gen11nv	⊢ ( 𝜑 ▶ ∀ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		gen11nv.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		gen11nv.2	⊢ ( 𝜑 ▶ 𝜓 )
3	2	in1	⊢ ( 𝜑 → 𝜓 )
4	1 3	alrimih	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
5	4	dfvd1ir	⊢ ( 𝜑 ▶ ∀ 𝑥 𝜓 )