Theorem syl5d 71

Description: A nested syllogism deduction. Deduction associated with syl5 33. (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 25	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜒)))
3		syl5d.2	. 2 ⊢ (𝜑 → (𝜃 → (𝜒 → 𝜏)))
4	2, 3	syldd 70	1 ⊢ (𝜑 → (𝜃 → (𝜓 → 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:  syl7  72  syl9  75  imim12d  79  sbi1  2380  mopick  2523  isofrlem  6490  kmlem9  8863  squeeze0  10805  lcmfunsnlem1  15188  fgss2  21488  ordcmp  31616  linepsubN  34056  pmapsub  34072  bgoldbnnsum3prm  40220