Theorem syldd 73

Description: Nested syllogism deduction. Deduction associated with syld 48. Double deduction associated with syl 18. (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 59	. 2 ⊢ ((𝜃 → 𝜏) → ((𝜒 → 𝜃) → (𝜒 → 𝜏)))
4	1, 2, 3	syl6c 71	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  74  syl6d  76  syl10  80  fvf1pr  7295  tfinds  7844  soseq  8143  tz7.49  8420  php3  9181  dffi2  9371  ordiso2  9465  rankuni2b  9813  oddprmdvds  16953  brbtwn2  29164  bj-exalims  37102  prtlem60  39489  lvoli2  40217