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 )