Theorem e333 45659

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

Hypotheses

Ref	Expression
e333.1	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
e333.2	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
e333.3	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
e333.4	⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))

Assertion

Ref	Expression
e333	⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Proof of Theorem e333

Step	Hyp	Ref	Expression
1		e333.3	. . . . . . 7 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
2	1	dfvd3i 45519	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
3	2	3imp 1128	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)
4		e333.1	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
5	4	dfvd3i 45519	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
6	5	3imp 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
7		e333.2	. . . . . . . . 9 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
8	7	dfvd3i 45519	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
9	8	3imp 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
10		e333.4	. . . . . . 7 ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))
11	6, 9, 10	syl2im 41	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜂 → 𝜁)))
12	11	pm2.43i 53	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜂 → 𝜁))
13	3, 12	syl5com 32	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜁))
14	13	pm2.43i 53	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜁)
15	14	3exp 1137	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜁)))
16	15	dfvd3ir 45520	1 ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103  (   wvd3 45514

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-vd3 45517

This theorem is used by:  e33  45660  e123  45688