Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfvd3 Structured version   Visualization version   GIF version

Theorem dfvd3 39332
Description: Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dfvd3 ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Proof of Theorem dfvd3
StepHypRef Expression
1 df-vd3 39331 . 2 ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ ((𝜑𝜓𝜒) → 𝜃))
2 df-3an 1073 . . . . 5 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
32imbi1i 338 . . . 4 (((𝜑𝜓𝜒) → 𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) → 𝜃))
4 impexp 437 . . . 4 ((((𝜑𝜓) ∧ 𝜒) → 𝜃) ↔ ((𝜑𝜓) → (𝜒𝜃)))
53, 4bitri 264 . . 3 (((𝜑𝜓𝜒) → 𝜃) ↔ ((𝜑𝜓) → (𝜒𝜃)))
6 impexp 437 . . 3 (((𝜑𝜓) → (𝜒𝜃)) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
75, 6bitri 264 . 2 (((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
81, 7bitri 264 1 ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  (   wvd3 39328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1073  df-vd3 39331
This theorem is referenced by:  dfvd3i  39333  dfvd3ir  39334
  Copyright terms: Public domain W3C validator