This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem e333

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

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

Proof

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