Theorem gen21nv 42240

Description: Virtual deduction form of alrimdh 1866. (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 42205	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	1, 2, 4	alrimdh 1866	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
6	5	dfvd2ir 42206	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

Syntax hints:   → wi 4  ∀wal 1537  (   wvd2 42197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-vd2 42198

This theorem is referenced by:  ssralv2VD  42486