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Theorem dfbi3 863
 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) ↔ ((φ ψ) φ ¬ ψ)))

Proof of Theorem dfbi3
StepHypRef Expression
1 xor 861 . 2 (¬ (φ ↔ ¬ ψ) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ)))
2 pm5.18 345 . 2 ((φψ) ↔ ¬ (φ ↔ ¬ ψ))
3 notnot 282 . . . 4 (ψ ↔ ¬ ¬ ψ)
43anbi2i 675 . . 3 ((φ ψ) ↔ (φ ¬ ¬ ψ))
5 ancom 437 . . 3 ((¬ φ ¬ ψ) ↔ (¬ ψ ¬ φ))
64, 5orbi12i 507 . 2 (((φ ψ) φ ¬ ψ)) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ)))
71, 2, 63bitr4i 268 1 ((φψ) ↔ ((φ ψ) φ ¬ ψ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by:  pm5.24  864  4exmid  905  nanbi  1294  ifbi  3679
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