Theorem gen11nv 45153

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1843 is gen11nv 45153 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
gen11nv.1	⊢ (𝜑 → ∀𝑥𝜑)
gen11nv.2	⊢ (   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
gen11nv	⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Proof of Theorem gen11nv

Step	Hyp	Ref	Expression
1		gen11nv.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		gen11nv.2	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
3	2	in1 45107	. . 3 ⊢ (𝜑 → 𝜓)
4	1, 3	alrimih 1843	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
5	4	dfvd1ir 45109	1 ⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Syntax hints:   → wi 4  ∀wal 1557  (   wvd1 45105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-vd1 45106

This theorem is referenced by:  tratrbVD  45396  hbimpgVD  45439  hbalgVD  45440  hbexgVD  45441  e2ebindVD  45447