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Metamath Proof Explorer

Theorem e112

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e112.1 ⊢ (    𝜑    ▶    𝜓    )
e112.2 ⊢ (    𝜑    ▶    𝜒    )
e112.3 ⊢ (    𝜑    ,    𝜃    ▶    𝜏    )
e112.4 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜏 → 𝜂 ) ) )
Assertion e112 (    𝜑    ,    𝜃    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e112.1 ⊢ (    𝜑    ▶    𝜓    )
2 e112.2 ⊢ (    𝜑    ▶    𝜒    )
3 e112.3 ⊢ (    𝜑    ,    𝜃    ▶    𝜏    )
4 e112.4 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜏 → 𝜂 ) ) )
5 1 vd12 ⊢ (    𝜑    ,    𝜃    ▶    𝜓    )
6 2 vd12 ⊢ (    𝜑    ,    𝜃    ▶    𝜒    )
7 5 6 3 4 e222 ⊢ (    𝜑    ,    𝜃    ▶    𝜂    )