MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm4.83 Structured version   Visualization version   GIF version

Theorem pm4.83 1018
Description: Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
Assertion
Ref Expression
pm4.83 (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓)

Proof of Theorem pm4.83
StepHypRef Expression
1 exmid 888 . . 3 (𝜑 ∨ ¬ 𝜑)
21a1bi 364 . 2 (𝜓 ↔ ((𝜑 ∨ ¬ 𝜑) → 𝜓))
3 jaob 955 . 2 (((𝜑 ∨ ¬ 𝜑) → 𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜓)))
42, 3bitr2i 277 1 (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841
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 208  df-an 397  df-or 842
This theorem is referenced by:  cases2  1039  dmdbr5ati  30126  cvlsupr3  36360  rp-fakeanorass  39757
  Copyright terms: Public domain W3C validator