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Theorem dfbi3 896
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  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )

Proof of Theorem dfbi3
StepHypRef Expression
1 xor 894 . 2  |-  ( -.  ( ph  <->  -.  ps )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) )
2 pm5.18 356 . 2  |-  ( (
ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) )
3 notnot 291 . . . 4  |-  ( ps  <->  -. 
-.  ps )
43anbi2i 694 . . 3  |-  ( (
ph  /\  ps )  <->  (
ph  /\  -.  -.  ps ) )
5 ancom 450 . . 3  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( -.  ps  /\  -.  ph ) )
64, 5orbi12i 521 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  <->  ( ( ph  /\ 
-.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) )
71, 2, 63bitr4i 279 1  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 186    \/ wo 368    /\ wa 369
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 187  df-or 370  df-an 371
This theorem is referenced by:  pm5.24  897  4exmid  942  nanbi  1356  nanbiOLD  1357  nanbiOLDOLD  1358  ifbi  3908  sqf11  23796  bj-dfbi4  30733  raaan2  37561  2reu4a  37575
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