Theorem syl3anl2

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005) (Proof shortened by Wolf Lammen, 27-Jun-2022)

	Ref	Expression
Hypotheses	syl3anl2.1	⊢ ( 𝜑 → 𝜒 )
	syl3anl2.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
Assertion	syl3anl2	⊢ ( ( ( 𝜓 ∧ 𝜑 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl3anl2.1	⊢ ( 𝜑 → 𝜒 )
2		syl3anl2.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
3	1	3anim2i	⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜃 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4	3 2	sylan	⊢ ( ( ( 𝜓 ∧ 𝜑 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )