Theorem e13an 44747

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

Hypotheses

Ref	Expression
e13an.1	⊢ (   𝜑   ▶   𝜓   )
e13an.2	⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )
e13an.3	⊢ ((𝜓 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
e13an	⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Proof of Theorem e13an

Step	Hyp	Ref	Expression
1		e13an.1	. 2 ⊢ (   𝜑   ▶   𝜓   )
2		e13an.2	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )
3		e13an.3	. . 3 ⊢ ((𝜓 ∧ 𝜏) → 𝜂)
4	3	ex 412	. 2 ⊢ (𝜓 → (𝜏 → 𝜂))
5	1, 2, 4	e13 44746	1 ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 395  (   wvd1 44567  (   wvd3 44585

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 207  df-an 396  df-3an 1089  df-vd1 44568  df-vd3 44588

This theorem is referenced by: (None)