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	$$⊢\left(\phi \to \psi \right)$$
Assertion	gen11	$$⊢\left(\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		gen11.1	$$⊢\left(\phi \to \psi \right)$$
2		dfvd1imp	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \psi \right)$$
3	1 2	ax-mp	$$⊢\phi \to \psi$$
4	3	alrimiv	$$⊢\phi \to \forall x\psi$$
5		dfvd1impr	$$⊢\left(\phi \to \forall x\psi \right)\to \left(\phi \to \forall x\psi \right)$$
6	4 5	ax-mp	$$⊢\left(\phi \to \forall x\psi \right)$$