Theorem syldd

Description: Nested syllogism deduction. Deduction associated with syld . Double deduction associated with syl . (Contributed by NM, 12-Dec-2004) (Proof shortened by Wolf Lammen, 11-May-2013)

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

Proof

Step	Hyp	Ref	Expression
1		syldd.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
2		syldd.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜏 ) ) )
3		imim2	⊢ ( ( 𝜃 → 𝜏 ) → ( ( 𝜒 → 𝜃 ) → ( 𝜒 → 𝜏 ) ) )
4	2 1 3	syl6c	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )