Theorem e233 43361

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e233.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e233.2	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )
e233.3	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
e233.4	⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))

Assertion

Ref	Expression
e233	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )

Proof of Theorem e233

Step	Hyp	Ref	Expression
1		e233.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	dfvd2i 43181	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		e233.2	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )
4	3	dfvd3i 43188	. . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
5		e233.3	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
6	5	dfvd3i 43188	. . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
7		e233.4	. . 3 ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))
8	2, 4, 6, 7	ee233 43115	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))
9	8	dfvd3ir 43189	1 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )

Syntax hints:   → wi 4  (   wvd2 43173  (   wvd3 43183

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 1089  df-vd2 43174  df-vd3 43186

This theorem is referenced by:  truniALTVD  43474  onfrALTlem2VD  43485