MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfbi3 Structured version   Visualization version   GIF version

Theorem dfbi3 1033
Description: An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (Proof shortened by NM, 29-Oct-2021.)
Assertion
Ref Expression
dfbi3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem dfbi3
StepHypRef Expression
1 con34b 308 . . 3 ((𝜓𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
21anbi2i 616 . 2 (((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
3 dfbi2 468 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
4 cases2 1031 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
52, 3, 43bitr4i 295 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836
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 199  df-an 387  df-or 837
This theorem is referenced by:  pm5.24  1034  nanbi  1569  ifbi  4328  sqf11  25321  bj-dfbi4  33140  raaan2  42104  2reu4a  42160
  Copyright terms: Public domain W3C validator