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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	$⊢ φ \to \forall x φ$
	gen11nv.2	$⊢ (φ \to ψ)$
Assertion	gen11nv	$⊢ (φ \to \forall x ψ)$

Proof

Step	Hyp	Ref	Expression
1		gen11nv.1	$⊢ φ \to \forall x φ$
2		gen11nv.2	$⊢ (φ \to ψ)$
3	2	in1	$⊢ φ \to ψ$
4	1 3	alrimih	$⊢ φ \to \forall x ψ$
5	4	dfvd1ir	$⊢ (φ \to \forall x ψ)$