Theorem e31 45519

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e31.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e31.2	⊢ (   𝜑   ▶   𝜏   )
e31.3	⊢ (𝜃 → (𝜏 → 𝜂))

Assertion

Ref	Expression
e31	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Proof of Theorem e31

Step	Hyp	Ref	Expression
1		e31.1	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
2		e31.2	. . 3 ⊢ (   𝜑   ▶   𝜏   )
3	2	vd13 45370	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
4		e31.3	. 2 ⊢ (𝜃 → (𝜏 → 𝜂))
5	1, 3, 4	e33 45502	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45338  (   wvd3 45356

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-vd1 45339  df-vd3 45359

This theorem is used by:  e31an  45521  trintALTVD  45648