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Theorem dfbi3 1818
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 1816 . 2 (¬ (φ ↔ ¬ ψ) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ)))
2 pm5.18 1817 . 2 ((φψ) ↔ ¬ (φ ↔ ¬ ψ))
3 notnot 744 . . . 4 (ψ ↔ ¬ ¬ ψ)
43anbi2i 429 . . 3 ((φ ψ) ↔ (φ ¬ ¬ ψ))
5 ancom 251 . . 3 ((¬ φ ¬ ψ) ↔ (¬ ψ ¬ φ))
64, 5orbi12i 656 . 2 (((φ ψ) φ ¬ ψ)) ↔ ((φ ¬ ¬ ψ) ψ ¬ φ)))
71, 2, 63bitr4i 199 1 ((φψ) ↔ ((φ ψ) φ ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 95  wb 96   wo 605
This theorem is referenced by:  pm5.24  1819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in1 526  ax-in2 527  ax-io 606  ax-3 714
This theorem depends on definitions:  df-bi 108
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