Theorem syl3anr3

Description: A syllogism inference. (Contributed by NM, 23-Aug-2007)

	Ref	Expression
Hypotheses	syl3anr3.1	⊢ ( 𝜑 → 𝜏 )
	syl3anr3.2	⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
Assertion	syl3anr3	⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜑 ) ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl3anr3.1	⊢ ( 𝜑 → 𝜏 )
2		syl3anr3.2	⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
3	1	3anim3i	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜑 ) → ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) )
4	3 2	sylan2	⊢ ( ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜑 ) ) → 𝜂 )