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Metamath Proof Explorer

Theorem syl3an1

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995)

Ref Expression
Hypotheses syl3an1.1 ⊢ ( 𝜑 → 𝜓 )
syl3an1.2 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion syl3an1 ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl3an1.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3an1.2 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
3 1 3anim1i ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4 3 2 syl ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )