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Metamath Proof Explorer

Theorem e012

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

Ref Expression
Hypotheses e012.1 ⊢ 𝜑
e012.2 ⊢ (    𝜓    ▶    𝜒    )
e012.3 ⊢ (    𝜓    ,    𝜃    ▶    𝜏    )
e012.4 ⊢ ( 𝜑 → ( 𝜒 → ( 𝜏 → 𝜂 ) ) )
Assertion e012 (    𝜓    ,    𝜃    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e012.1 ⊢ 𝜑
2 e012.2 ⊢ (    𝜓    ▶    𝜒    )
3 e012.3 ⊢ (    𝜓    ,    𝜃    ▶    𝜏    )
4 e012.4 ⊢ ( 𝜑 → ( 𝜒 → ( 𝜏 → 𝜂 ) ) )
5 1 vd01 ⊢ (    𝜓    ▶    𝜑    )
6 5 2 3 4 e112 ⊢ (    𝜓    ,    𝜃    ▶    𝜂    )