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Metamath Proof Explorer

Theorem syl3an

Description: A triple syllogism inference. (Contributed by NM, 13-May-2004)

Ref Expression
Hypotheses syl3an.1 ⊢ ( 𝜑 → 𝜓 )
syl3an.2 ⊢ ( 𝜒 → 𝜃 )
syl3an.3 ⊢ ( 𝜏 → 𝜂 )
syl3an.4 ⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
Assertion syl3an ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )

Proof

Step Hyp Ref Expression
1 syl3an.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3an.2 ⊢ ( 𝜒 → 𝜃 )
3 syl3an.3 ⊢ ( 𝜏 → 𝜂 )
4 syl3an.4 ⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
5 1 2 3 3anim123i ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) )
6 5 4 syl ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )