Theorem syldd 72

Description: Nested syllogism deduction. Deduction associated with syld 47. 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 58	. 2 ⊢ ((𝜃 → 𝜏) → ((𝜒 → 𝜃) → (𝜒 → 𝜏)))
4	1, 2, 3	syl6c 70	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  73  syl6d  75  syl10  79  fvf1pr  7343  tfinds  7897  soseq  8200  tz7.49  8501  php3  9275  dffi2  9492  ordiso2  9584  rankuni2b  9922  oddprmdvds  16950  brbtwn2  28938  bj-exalims  36600  prtlem60  38809  lvoli2  39538