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Theorem 4exmid 905
Description: The disjunction of the four possible combinations of two wffs and their negations is always true. (Contributed by David Abernethy, 28-Jan-2014.)
Assertion
Ref Expression
4exmid  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  \/  ( (
ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) )

Proof of Theorem 4exmid
StepHypRef Expression
1 exmid 404 . 2  |-  ( (
ph 
<->  ps )  \/  -.  ( ph  <->  ps ) )
2 dfbi3 863 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
3 xor 861 . . 3  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
42, 3orbi12i 507 . 2  |-  ( ( ( ph  <->  ps )  \/  -.  ( ph  <->  ps )
)  <->  ( ( (
ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  \/  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) ) )
51, 4mpbi 199 1  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  \/  ( (
ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) )
Colors of variables: wff set 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
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