MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.17 Structured version   Visualization version   GIF version

Theorem pm5.17 950
Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
Assertion
Ref Expression
pm5.17 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.17
StepHypRef Expression
1 bicom 212 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝜑))
2 dfbi2 661 . 2 ((¬ 𝜓𝜑) ↔ ((¬ 𝜓𝜑) ∧ (𝜑 → ¬ 𝜓)))
3 orcom 401 . . . 4 ((𝜑𝜓) ↔ (𝜓𝜑))
4 df-or 384 . . . 4 ((𝜓𝜑) ↔ (¬ 𝜓𝜑))
53, 4bitr2i 265 . . 3 ((¬ 𝜓𝜑) ↔ (𝜑𝜓))
6 imnan 437 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
75, 6anbi12i 733 . 2 (((¬ 𝜓𝜑) ∧ (𝜑 → ¬ 𝜓)) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
81, 2, 73bitrri 287 1 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383
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-or 384  df-an 385
This theorem is referenced by:  nbi2  954  odd2np1  15112  ordtconnlem1  30098  sgnneg  30730
  Copyright terms: Public domain W3C validator