Theorem syland

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004)

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

Proof

Step	Hyp	Ref	Expression
1		syland.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syland.2	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜏 ) )
3	2	expd	⊢ ( 𝜑 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
4	1 3	syld	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜏 ) ) )
5	4	impd	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → 𝜏 ) )