Theorem e123 45585

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e123.1	⊢ (   𝜑   ▶   𝜓   )
e123.2	⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
e123.3	⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜂   )
e123.4	⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))

Assertion

Ref	Expression
e123	⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )

Proof of Theorem e123

Step	Hyp	Ref	Expression
1		e123.1	. . 3 ⊢ (   𝜑   ▶   𝜓   )
2	1	vd13 45425	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜓   )
3		e123.2	. . 3 ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
4	3	vd23 45426	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜃   )
5		e123.3	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜂   )
6		e123.4	. 2 ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))
7	2, 4, 5, 6	e333 45556	1 ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45393  (   wvd2 45401  (   wvd3 45411

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 45394  df-vd2 45402  df-vd3 45414

This theorem is used by:  suctrALT2VD  45659