Theorem syldd 67

Description: Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)

Hypotheses

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

Assertion

Ref	Expression
syldd	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Proof of Theorem syldd

Step	Hyp	Ref	Expression
1		syldd.2	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
2		syldd.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3		imim2 55	. 2 ⊢ ((𝜃 → 𝜏) → ((𝜒 → 𝜃) → (𝜒 → 𝜏)))
4	1, 2, 3	syl6c 66	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  syl5d  68  syl6d  70  syl10  1484  ordiso2  7376  oddprmdvds  13156