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Metamath Proof Explorer

Theorem syl3anr2

Description: A syllogism inference. (Contributed by NM, 1-Aug-2007) (Proof shortened by Wolf Lammen, 27-Jun-2022)

Ref Expression
Hypotheses syl3anr2.1 ⊢ ( 𝜑 → 𝜃 )
syl3anr2.2 ⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
Assertion syl3anr2 ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜑 ∧ 𝜏 ) ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl3anr2.1 ⊢ ( 𝜑 → 𝜃 )
2 syl3anr2.2 ⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
3 1 3anim2i ⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜏 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) )
4 3 2 sylan2 ⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜑 ∧ 𝜏 ) ) → 𝜂 )