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Metamath Proof Explorer

Theorem syl3an2

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995) (Proof shortened by Wolf Lammen, 26-Jun-2022)

Ref Expression
Hypotheses syl3an2.1 ⊢ ( 𝜑 → 𝜒 )
syl3an2.2 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion syl3an2 ( ( 𝜓 ∧ 𝜑 ∧ 𝜃 ) → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl3an2.1 ⊢ ( 𝜑 → 𝜒 )
2 syl3an2.2 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
3 1 3anim2i ⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜃 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4 3 2 syl ⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜃 ) → 𝜏 )