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Metamath Proof Explorer

Theorem syl3an12

Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024)

Ref Expression
Hypotheses syl3an12.1 ⊢ ( 𝜑 → 𝜓 )
syl3an12.2 ⊢ ( 𝜒 → 𝜃 )
syl3an12.s ⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion syl3an12 ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl3an12.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3an12.2 ⊢ ( 𝜒 → 𝜃 )
3 syl3an12.s ⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 )
4 id ⊢ ( 𝜏 → 𝜏 )
5 1 2 4 3 syl3an ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )