Theorem e33 45715

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e33.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e33.2	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
e33.3	⊢ (𝜃 → (𝜏 → 𝜂))

Assertion

Ref	Expression
e33	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Proof of Theorem e33

Step	Hyp	Ref	Expression
1		e33.1	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
2		e33.2	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
3		e33.3	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
4	3	a1i 11	. 2 ⊢ (𝜃 → (𝜃 → (𝜏 → 𝜂)))
5	1, 1, 2, 4	e333 45714	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

This proof depends on syntax axioms:   → wi 4  (   wvd3 45569

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

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-vd3 45572

This theorem is used by:  e33an  45716  e3  45718  e03  45721  e30  45725  e13  45729  e31  45732  e23  45736  e32  45739  truniALTVD  45859  trintALTVD  45861  onfrALTlem2VD  45870