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Metamath Proof Explorer

Theorem syld3an3

Description: A syllogism inference. (Contributed by NM, 20-May-2007)

Ref Expression
Hypotheses syld3an3.1 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
syld3an3.2 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion syld3an3 ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 syld3an3.1 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
2 syld3an3.2 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )
3 simp1 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜑 )
4 simp2 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜓 )
5 3 4 1 2 syl3anc ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )