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Metamath Proof Explorer

Theorem e222

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

Ref Expression
Hypotheses e222.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
e222.2 ⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
e222.3 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
e222.4 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
Assertion e222 (    𝜑    ,    𝜓    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e222.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
2 e222.2 ⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
3 e222.3 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
4 e222.4 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
5 3 dfvd2i ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
6 5 imp ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜏 )
7 1 dfvd2i ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
8 7 imp ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
9 2 dfvd2i ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
10 9 imp ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
11 8 10 4 syl2im ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ) → ( 𝜏 → 𝜂 ) ) )
12 11 pm2.43i ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜏 → 𝜂 ) )
13 6 12 syl5com ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ) → 𝜂 ) )
14 13 pm2.43i ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜂 )
15 14 ex ⊢ ( 𝜑 → ( 𝜓 → 𝜂 ) )
16 15 dfvd2ir ⊢ (    𝜑    ,    𝜓    ▶    𝜂    )