Theorem sylanr1

Description: A syllogism inference. (Contributed by NM, 9-Apr-2005)

	Ref	Expression
Hypotheses	sylanr1.1	⊢ ( 𝜑 → 𝜒 )
	sylanr1.2	⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜏 )
Assertion	sylanr1	⊢ ( ( 𝜓 ∧ ( 𝜑 ∧ 𝜃 ) ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		sylanr1.1	⊢ ( 𝜑 → 𝜒 )
2		sylanr1.2	⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜏 )
3	1	anim1i	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜃 ) )
4	3 2	sylan2	⊢ ( ( 𝜓 ∧ ( 𝜑 ∧ 𝜃 ) ) → 𝜏 )