Theorem syl3anl3

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

	Ref	Expression
Hypotheses	syl3anl3.1	⊢ ( 𝜑 → 𝜃 )
	syl3anl3.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
Assertion	syl3anl3	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜑 ) ∧ 𝜏 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl3anl3.1	⊢ ( 𝜑 → 𝜃 )
2		syl3anl3.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
3	1	3anim3i	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜑 ) → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
4	3 2	sylan	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜑 ) ∧ 𝜏 ) → 𝜂 )