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Theorem dfbi3 864
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 862 . 2  |-  ( -.  ( ph  <->  -.  ps )  <->  ( ( ph  /\  -.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) )
2 pm5.18 346 . 2  |-  ( (
ph 
<->  ps )  <->  -.  ( ph 
<->  -.  ps ) )
3 notnot 283 . . . 4  |-  ( ps  <->  -. 
-.  ps )
43anbi2i 676 . . 3  |-  ( (
ph  /\  ps )  <->  (
ph  /\  -.  -.  ps ) )
5 ancom 438 . . 3  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( -.  ps  /\  -.  ph ) )
64, 5orbi12i 508 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  <->  ( ( ph  /\ 
-.  -.  ps )  \/  ( -.  ps  /\  -.  ph ) ) )
71, 2, 63bitr4i 269 1  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359
This theorem is referenced by:  pm5.24  865  4exmid  906  nanbi  1303  ifbi  3748  sqf11  20910  raaan2  27867  2reu4a  27881
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 178  df-or 360  df-an 361
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