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Metamath Proof Explorer

Theorem e212

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

Ref Expression
Hypotheses e212.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
e212.2 ⊢ (    𝜑    ▶    𝜃    )
e212.3 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
e212.4 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
Assertion e212 (    𝜑    ,    𝜓    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e212.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
2 e212.2 ⊢ (    𝜑    ▶    𝜃    )
3 e212.3 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
4 e212.4 ⊢ ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
5 2 vd12 ⊢ (    𝜑    ,    𝜓    ▶    𝜃    )
6 1 5 3 4 e222 ⊢ (    𝜑    ,    𝜓    ▶    𝜂    )