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Metamath Proof Explorer

Theorem e12an

Description: Conjunction form of e12 (see syl6an ). (Contributed by Alan Sare, 11-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e12an.1 ⊢ (    𝜑    ▶    𝜓    )
e12an.2 ⊢ (    𝜑    ,    𝜒    ▶    𝜃    )
e12an.3 ⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion e12an (    𝜑    ,    𝜒    ▶    𝜏    )

Proof

Step Hyp Ref Expression
1 e12an.1 ⊢ (    𝜑    ▶    𝜓    )
2 e12an.2 ⊢ (    𝜑    ,    𝜒    ▶    𝜃    )
3 e12an.3 ⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
4 3 ex ⊢ ( 𝜓 → ( 𝜃 → 𝜏 ) )
5 1 2 4 e12 ⊢ (    𝜑    ,    𝜒    ▶    𝜏    )