Theorem syldd 70

Description: Nested syllogism deduction. Deduction associated with syld 46. Double deduction associated with syl 17. (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 56	. 2 ⊢ ((𝜃 → 𝜏) → ((𝜒 → 𝜃) → (𝜒 → 𝜏)))
4	1, 2, 3	syl6c 68	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  syl5d  71  syl6d  73  syl10  77  tfinds  6951  tz7.49  7427  dffi2  8212  ordiso2  8303  rankuni2b  8599  oddprmdvds  15445  brbtwn2  25585  soseq  30995  prtlem60  33152  lvoli2  33885