Theorem gen11nv 42126

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1827 is gen11nv 42126 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 42080	. . 3 ⊢ (𝜑 → 𝜓)
4	1, 3	alrimih 1827	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
5	4	dfvd1ir 42082	1 ⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Syntax hints:   → wi 4  ∀wal 1537  (   wvd1 42078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-vd1 42079

This theorem is referenced by:  tratrbVD  42370  hbimpgVD  42413  hbalgVD  42414  hbexgVD  42415  e2ebindVD  42421