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Metamath Proof Explorer

Theorem e111

Description: A virtual deduction elimination rule (see syl3c ). (Contributed by Alan Sare, 14-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e111.1 ⊢ (    𝜑    ▶    𝜓    )
e111.2 ⊢ (    𝜑    ▶    𝜒    )
e111.3 ⊢ (    𝜑    ▶    𝜃    )
e111.4 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
Assertion e111 (    𝜑    ▶    𝜏    )

Proof

Step Hyp Ref Expression
1 e111.1 ⊢ (    𝜑    ▶    𝜓    )
2 e111.2 ⊢ (    𝜑    ▶    𝜒    )
3 e111.3 ⊢ (    𝜑    ▶    𝜃    )
4 e111.4 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
5 3 in1 ⊢ ( 𝜑 → 𝜃 )
6 1 in1 ⊢ ( 𝜑 → 𝜓 )
7 2 in1 ⊢ ( 𝜑 → 𝜒 )
8 6 7 4 syl2im ⊢ ( 𝜑 → ( 𝜑 → ( 𝜃 → 𝜏 ) ) )
9 8 pm2.43i ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
10 5 9 syl5com ⊢ ( 𝜑 → ( 𝜑 → 𝜏 ) )
11 10 pm2.43i ⊢ ( 𝜑 → 𝜏 )
12 11 dfvd1ir ⊢ (    𝜑    ▶    𝜏    )