Theorem syl6d 76

Description: A nested syllogism deduction. Deduction associated with syl6 36. (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 26	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
4	1, 3	syldd 73	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:  syl8  77  omlimcl  8569  ltexprlem7  11055  axpre-sup  11182  caubnd  15450  ubthlem1  31359  axuntco  37106  poimirlem29  38406  ee13  45335  ssralv2  45362  rspsbc2  45365  truniALT  45372  stgoldbwt  48700