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	$$⊢\phi \to \forall x\phi$$
	gen11nv.2	$$⊢\left(\phi \to \psi \right)$$
Assertion	gen11nv	$$⊢\left(\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		gen11nv.1	$$⊢\phi \to \forall x\phi$$
2		gen11nv.2	$$⊢\left(\phi \to \psi \right)$$
3	2	in1	$$⊢\phi \to \psi$$
4	1 3	alrimih	$$⊢\phi \to \forall x\psi$$
5	4	dfvd1ir	$$⊢\left(\phi \to \forall x\psi \right)$$