Theorem e112 45478

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

Hypotheses

Ref	Expression
e112.1	⊢ (   𝜑   ▶   𝜓   )
e112.2	⊢ (   𝜑   ▶   𝜒   )
e112.3	⊢ (   𝜑   ,   𝜃   ▶   𝜏   )
e112.4	⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂)))

Assertion

Ref	Expression
e112	⊢ (   𝜑   ,   𝜃   ▶   𝜂   )

Proof of Theorem e112

Step	Hyp	Ref	Expression
1		e112.1	. . 3 ⊢ (   𝜑   ▶   𝜓   )
2	1	vd12 45424	. 2 ⊢ (   𝜑   ,   𝜃   ▶   𝜓   )
3		e112.2	. . 3 ⊢ (   𝜑   ▶   𝜒   )
4	3	vd12 45424	. 2 ⊢ (   𝜑   ,   𝜃   ▶   𝜒   )
5		e112.3	. 2 ⊢ (   𝜑   ,   𝜃   ▶   𝜏   )
6		e112.4	. 2 ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂)))
7	2, 4, 5, 6	e222 45460	1 ⊢ (   𝜑   ,   𝜃   ▶   𝜂   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45393  (   wvd2 45401

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-vd1 45394  df-vd2 45402

This theorem is used by:  e012  45491  e102  45493