Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbc3orgOLD Structured version   Visualization version   GIF version

Theorem sbc3orgOLD 38568
Description: sbcorgOLD 38566 with a 3-disjuncts. This proof is sbc3orgVD 38912 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbc3orgOLD (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

Proof of Theorem sbc3orgOLD
StepHypRef Expression
1 sbcorgOLD 38566 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)))
2 df-3or 1038 . . . . 5 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
32bicomi 214 . . . 4 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
43sbcbiiOLD 38567 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒)))
5 sbcorgOLD 38566 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
65orbi1d 739 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)))
71, 4, 63bitr3d 298 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)))
8 df-3or 1038 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))
97, 8syl6bbr 278 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  w3o 1036  wcel 1989  [wsbc 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3200  df-sbc 3434
This theorem is referenced by:  sbcoreleleqVD  38921
  Copyright terms: Public domain W3C validator