Theorem e123 42382

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 42221	. 2 ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜓   )
3		e123.2	. . 3 ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
4	3	vd23 42222	. 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 42353	1 ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )

Syntax hints:   → wi 4  (   wvd1 42189  (   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-vd1 42190  df-vd2 42198  df-vd3 42210

This theorem is referenced by:  suctrALT2VD  42456