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Theorem dfbi3 1045
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 319 . . 3 ((𝜓𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
21anbi2i 625 . 2 (((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
3 dfbi2 478 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
4 cases2 1043 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
52, 3, 43bitr4i 306 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844
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 210  df-an 400  df-or 845
This theorem is referenced by:  pm5.24  1046  nanbi  1491  raaan2  4422  2reu4lem  4423  ifbi  4446  sqf11  25724  bj-dfbi4  34019
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