Theorem syl3anr1

Description: A syllogism inference. (Contributed by NM, 31-Jul-2007)

	Ref	Expression
Hypotheses	syl3anr1.1	⊢ ( 𝜑 → 𝜓 )
	syl3anr1.2	⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
Assertion	syl3anr1	⊢ ( ( 𝜒 ∧ ( 𝜑 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl3anr1.1	⊢ ( 𝜑 → 𝜓 )
2		syl3anr1.2	⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
3	1	3anim1i	⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜏 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) )
4	3 2	sylan2	⊢ ( ( 𝜒 ∧ ( 𝜑 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )