Theorem gen31 45740

Description: Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 45740 is ggen31 45664 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen31.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )

Assertion

Ref	Expression
gen31	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   ∀𝑥𝜃   )

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥   𝜓,𝑥

Allowed substitution hint:   𝜃(𝑥)

Proof of Theorem gen31

Step	Hyp	Ref	Expression
1		gen31.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
2	1	dfvd3i 45711	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	ggen31 45664	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))
4	3	dfvd3ir 45712	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   ∀𝑥𝜃   )

This proof depends on syntax axioms:  ∀wal 1567  (   wvd3 45706

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-vd3 45709

This theorem is used by:  onfrALTlem2VD  46007