Theorem syl6d 75

Description: A nested syllogism deduction. Deduction associated with syl6 35. (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)

Hypotheses

Ref	Expression
syl6d.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
syl6d.2	⊢ (𝜑 → (𝜃 → 𝜏))

Assertion

Ref	Expression
syl6d	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Proof of Theorem syl6d

Step	Hyp	Ref	Expression
1		syl6d.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2		syl6d.2	. . 3 ⊢ (𝜑 → (𝜃 → 𝜏))
3	2	a1d 25	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
4	1, 3	syldd 72	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:  syl8  76  omlimcl  8371  ltexprlem7  10729  axpre-sup  10856  caubnd  14998  ubthlem1  29133  poimirlem29  35733  ee13  42013  ssralv2  42040  rspsbc2  42043  truniALT  42050  stgoldbwt  45116