Theorem gen21nv 45357

Description: Virtual deduction form of alrimdh 1892. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
gen21nv.1	⊢ (𝜑 → ∀𝑥𝜑)
gen21nv.2	⊢ (𝜓 → ∀𝑥𝜓)
gen21nv.3	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )

Assertion

Ref	Expression
gen21nv	⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

Proof of Theorem gen21nv

Step	Hyp	Ref	Expression
1		gen21nv.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		gen21nv.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3		gen21nv.3	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
4	3	dfvd2i 45322	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	1, 2, 4	alrimdh 1892	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
6	5	dfvd2ir 45323	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  (   wvd2 45314

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-an 401  df-vd2 45315

This theorem is used by:  ssralv2VD  45602