Theorem syldanl

Description: A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		syldanl.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		syldanl.2	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
3	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	3	imdistani	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) )
5	4 2	sylan	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) → 𝜏 )