Theorem sylanl1

Description: A syllogism inference. (Contributed by NM, 10-Mar-2005)

	Ref	Expression
Hypotheses	sylanl1.1	⊢ ( 𝜑 → 𝜓 )
	sylanl1.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
Assertion	sylanl1	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		sylanl1.1	⊢ ( 𝜑 → 𝜓 )
2		sylanl1.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
3	1	anim1i	⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ∧ 𝜒 ) )
4	3 2	sylan	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )