Theorem sylanl2

Description: A syllogism inference. (Contributed by NM, 1-Jan-2005)

	Ref	Expression
Hypotheses	sylanl2.1	⊢ ( 𝜑 → 𝜒 )
	sylanl2.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
Assertion	sylanl2	⊢ ( ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		sylanl2.1	⊢ ( 𝜑 → 𝜒 )
2		sylanl2.2	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
3	1	adantl	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 )
4	3 2	syldanl	⊢ ( ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜃 ) → 𝜏 )