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Theorem nanbi 1294
Description: Show equivalence between the bidirectional and the Nicod version. (Contributed by Jeff Hoffman, 19-Nov-2007.)
Assertion
Ref Expression
nanbi  |-  ( (
ph 
<->  ps )  <->  ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) ) )

Proof of Theorem nanbi
StepHypRef Expression
1 pm4.57 483 . 2  |-  ( -.  ( -.  ( ph  /\ 
ps )  /\  -.  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
2 df-nan 1288 . . 3  |-  ( ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )  <->  -.  (
( ph  -/\  ps )  /\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) ) )
3 df-nan 1288 . . . 4  |-  ( (
ph  -/\  ps )  <->  -.  ( ph  /\  ps ) )
4 df-nan 1288 . . . . 5  |-  ( ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
)  <->  -.  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps ) ) )
5 nannot 1293 . . . . . 6  |-  ( -. 
ph 
<->  ( ph  -/\  ph )
)
6 nannot 1293 . . . . . 6  |-  ( -. 
ps 
<->  ( ps  -/\  ps )
)
75, 6anbi12i 678 . . . . 5  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps )
) )
84, 7xchbinxr 302 . . . 4  |-  ( ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
)  <->  -.  ( -.  ph 
/\  -.  ps )
)
93, 8anbi12i 678 . . 3  |-  ( ( ( ph  -/\  ps )  /\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )  <->  ( -.  ( ph  /\  ps )  /\  -.  ( -.  ph  /\ 
-.  ps ) ) )
102, 9xchbinx 301 . 2  |-  ( ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )  <->  -.  ( -.  ( ph  /\  ps )  /\  -.  ( -. 
ph  /\  -.  ps )
) )
11 dfbi3 863 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
121, 10, 113bitr4ri 269 1  |-  ( (
ph 
<->  ps )  <->  ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    -/\ wnan 1287
This theorem is referenced by:  nic-dfim  1424  nic-dfneg  1425
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  df-nan 1288
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