Theorem e31an 44186

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e31an.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e31an.2	⊢ (   𝜑   ▶   𝜏   )
e31an.3	⊢ ((𝜃 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
e31an	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Proof of Theorem e31an

Step	Hyp	Ref	Expression
1		e31an.1	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
2		e31an.2	. 2 ⊢ (   𝜑   ▶   𝜏   )
3		e31an.3	. . 3 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
4	3	ex 412	. 2 ⊢ (𝜃 → (𝜏 → 𝜂))
5	1, 2, 4	e31 44184	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 395  (   wvd1 44002  (   wvd3 44020

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

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-vd1 44003  df-vd3 44023

This theorem is referenced by: (None)