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Metamath Proof Explorer

Theorem e21an

Description: Conjunction form of e21 . (Contributed by Alan Sare, 15-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e21an.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
e21an.2 ⊢ (    𝜑    ▶    𝜃    )
e21an.3 ⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion e21an (    𝜑    ,    𝜓    ▶    𝜏    )

Proof

Step Hyp Ref Expression
1 e21an.1 ⊢ (    𝜑    ,    𝜓    ▶    𝜒    )
2 e21an.2 ⊢ (    𝜑    ▶    𝜃    )
3 e21an.3 ⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
4 3 ex ⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
5 1 2 4 e21 ⊢ (    𝜑    ,    𝜓    ▶    𝜏    )