Theorem syl3anl1

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005)

	Ref	Expression
Hypotheses	syl3anl1.1	⊢ ( 𝜑 → 𝜓 )
	syl3anl1.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
Assertion	syl3anl1	⊢ ( ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl3anl1.1	⊢ ( 𝜑 → 𝜓 )
2		syl3anl1.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
3	1	3anim1i	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4	3 2	sylan	⊢ ( ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )