Theorem syl5d 74

Description: A nested syllogism deduction. Deduction associated with syl5 35. (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)

Hypotheses

Ref	Expression
syl5d.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl5d.2	⊢ (𝜑 → (𝜃 → (𝜒 → 𝜏)))

Assertion

Ref	Expression
syl5d	⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))

Proof of Theorem syl5d

Step	Hyp	Ref	Expression
1		syl5d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	a1d 26	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜒)))
3		syl5d.2	. 2 ⊢ (𝜑 → (𝜃 → (𝜒 → 𝜏)))
4	2, 3	syldd 73	1 ⊢ (𝜑 → (𝜃 → (𝜓 → 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:  syl7  75  syl9  78  imim12d  82  mopick  2652  isofrlem  7345  kmlem9  10165  squeeze0  12146  lcmfunsnlem1  16733  rnglidlmcl  21410  fgss2  24106  ordcmp  37074  linepsubN  40633  pmapsub  40649  relpfrlem  45784  ichreuopeq  48381  bgoldbnnsum3prm  48728  uhgrimedgi  48814  grimedg  48859