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Metamath Proof Explorer

Theorem e121

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

Ref Expression
Hypotheses e121.1 ⊢ (    𝜑    ▶    𝜓    )
e121.2 ⊢ (    𝜑    ,    𝜒    ▶    𝜃    )
e121.3 ⊢ (    𝜑    ▶    𝜏    )
e121.4 ⊢ ( 𝜓 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
Assertion e121 (    𝜑    ,    𝜒    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e121.1 ⊢ (    𝜑    ▶    𝜓    )
2 e121.2 ⊢ (    𝜑    ,    𝜒    ▶    𝜃    )
3 e121.3 ⊢ (    𝜑    ▶    𝜏    )
4 e121.4 ⊢ ( 𝜓 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) )
5 1 vd12 ⊢ (    𝜑    ,    𝜒    ▶    𝜓    )
6 3 vd12 ⊢ (    𝜑    ,    𝜒    ▶    𝜏    )
7 5 2 6 4 e222 ⊢ (    𝜑    ,    𝜒    ▶    𝜂    )