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Metamath Proof Explorer

Theorem e33an

Description: Conjunction form of e33 . (Contributed by Alan Sare, 15-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e33an.1 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    )
e33an.2 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜏    )
e33an.3 ⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion e33an (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜂    )

Proof

Step Hyp Ref Expression
1 e33an.1 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜃    )
2 e33an.2 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜏    )
3 e33an.3 ⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
4 3 ex ⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) )
5 1 2 4 e33 ⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    𝜂    )