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Theorem nanbiOLDOLD 1394
Description: Obsolete proof of nanbi 1392 as of 8-Mar-2020. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nanbiOLDOLD  |-  ( (
ph 
<->  ps )  <->  ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) ) )

Proof of Theorem nanbiOLDOLD
StepHypRef Expression
1 pm4.57 500 . 2  |-  ( -.  ( -.  ( ph  /\ 
ps )  /\  -.  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
2 df-nan 1384 . . 3  |-  ( ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )  <->  -.  (
( ph  -/\  ps )  /\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) ) )
3 df-nan 1384 . . . 4  |-  ( (
ph  -/\  ps )  <->  -.  ( ph  /\  ps ) )
4 df-nan 1384 . . . . 5  |-  ( ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
)  <->  -.  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps ) ) )
5 nannot 1391 . . . . . 6  |-  ( -. 
ph 
<->  ( ph  -/\  ph )
)
6 nannot 1391 . . . . . 6  |-  ( -. 
ps 
<->  ( ps  -/\  ps )
)
75, 6anbi12i 702 . . . . 5  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps )
) )
84, 7xchbinxr 313 . . . 4  |-  ( ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
)  <->  -.  ( -.  ph 
/\  -.  ps )
)
93, 8anbi12i 702 . . 3  |-  ( ( ( ph  -/\  ps )  /\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )  <->  ( -.  ( ph  /\  ps )  /\  -.  ( -.  ph  /\ 
-.  ps ) ) )
102, 9xchbinx 312 . 2  |-  ( ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )  <->  -.  ( -.  ( ph  /\  ps )  /\  -.  ( -. 
ph  /\  -.  ps )
) )
11 dfbi3 903 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
121, 10, 113bitr4ri 282 1  |-  ( (
ph 
<->  ps )  <->  ( ( ph  -/\  ps )  -/\  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/ wo 370    /\ wa 371    -/\ wnan 1383
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 189  df-or 372  df-an 373  df-nan 1384
This theorem is referenced by: (None)
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