Theorem e23an 42376

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

Hypotheses

Ref	Expression
e23an.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e23an.2	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )
e23an.3	⊢ ((𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
e23an	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )

Proof of Theorem e23an

Step	Hyp	Ref	Expression
1		e23an.1	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2		e23an.2	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )
3		e23an.3	. . 3 ⊢ ((𝜒 ∧ 𝜏) → 𝜂)
4	3	ex 413	. 2 ⊢ (𝜒 → (𝜏 → 𝜂))
5	1, 2, 4	e23 42375	1 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 396  (   wvd2 42197  (   wvd3 42207

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 397  df-3an 1088  df-vd2 42198  df-vd3 42210

This theorem is referenced by: (None)