Theorem syl3an12

Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024)

	Ref	Expression
Hypotheses	syl3an12.1	⊢ ( 𝜑 → 𝜓 )
	syl3an12.2	⊢ ( 𝜒 → 𝜃 )
	syl3an12.s	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion	syl3an12	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl3an12.1	⊢ ( 𝜑 → 𝜓 )
2		syl3an12.2	⊢ ( 𝜒 → 𝜃 )
3		syl3an12.s	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) → 𝜂 )
4		id	⊢ ( 𝜏 → 𝜏 )
5	1 2 4 3	syl3an	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )