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Metamath Proof Explorer

Theorem e123

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e123.1 ⊢ (    𝜑    ▶    𝜓    )
e123.2 ⊢ (    𝜑    ,    𝜒    ▶    𝜃    )
e123.3 ⊢ (    𝜑    ,    𝜒    ,    𝜏    ▶    𝜂    )
e123.4 ⊢ ( 𝜓 → ( 𝜃 → ( 𝜂 → 𝜁 ) ) )
Assertion e123 (    𝜑    ,    𝜒    ,    𝜏    ▶    𝜁    )

Proof

Step Hyp Ref Expression
1 e123.1 ⊢ (    𝜑    ▶    𝜓    )
2 e123.2 ⊢ (    𝜑    ,    𝜒    ▶    𝜃    )
3 e123.3 ⊢ (    𝜑    ,    𝜒    ,    𝜏    ▶    𝜂    )
4 e123.4 ⊢ ( 𝜓 → ( 𝜃 → ( 𝜂 → 𝜁 ) ) )
5 1 vd13 ⊢ (    𝜑    ,    𝜒    ,    𝜏    ▶    𝜓    )
6 2 vd23 ⊢ (    𝜑    ,    𝜒    ,    𝜏    ▶    𝜃    )
7 5 6 3 4 e333 ⊢ (    𝜑    ,    𝜒    ,    𝜏    ▶    𝜁    )