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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  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )

Proof of Theorem dfbi3
StepHypRef Expression
1 xor 861 . 2  |-  ( -.  ( ph  <->  -.  ps )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) )
2 pm5.18 345 . 2  |-  ( (
ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) )
3 notnot 282 . . . 4  |-  ( ps  <->  -. 
-.  ps )
43anbi2i 675 . . 3  |-  ( (
ph  /\  ps )  <->  (
ph  /\  -.  -.  ps ) )
5 ancom 437 . . 3  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( -.  ps  /\  -.  ph ) )
64, 5orbi12i 507 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  <->  ( ( ph  /\ 
-.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) )
71, 2, 63bitr4i 268 1  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  pm5.24  864  4exmid  905  nanbi  1294  ifbi  3595  sqf11  20393  raaan2  28056  2reu4a  28070
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
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