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Theorem dfbi3 932
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.)
Assertion
Ref Expression
dfbi3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem dfbi3
StepHypRef Expression
1 xor 930 . 2 (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))
2 pm5.18 369 . 2 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
3 notnotb 302 . . . 4 (𝜓 ↔ ¬ ¬ 𝜓)
43anbi2i 725 . . 3 ((𝜑𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓))
5 ancom 464 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
64, 5orbi12i 541 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))
71, 2, 63bitr4i 290 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wo 381  wa 382
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 195  df-or 383  df-an 384
This theorem is referenced by:  pm5.24  933  4exmid  976  nanbi  1445  ifbi  4056  sqf11  24610  bj-dfbi4  31562  raaan2  39648  2reu4a  39662
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