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Metamath Proof Explorer

Theorem e32an

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

Ref Expression
Hypotheses e32an.1 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    )
e32an.2 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
e32an.3 ⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion e32an (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e32an.1 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    )
2 e32an.2 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )
3 e32an.3 ⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
4 3 ex ⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) )
5 1 2 4 e32 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜂    )