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Theorem abnotataxb 27988
Description: Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
Hypotheses
Ref Expression
abnotataxb.1  |-  -.  ph
abnotataxb.2  |-  ps
Assertion
Ref Expression
abnotataxb  |-  ( ph  \/_ 
ps )

Proof of Theorem abnotataxb
StepHypRef Expression
1 abnotataxb.1 . . . . . . 7  |-  -.  ph
2 abnotataxb.2 . . . . . . 7  |-  ps
31, 2pm3.2i 441 . . . . . 6  |-  ( -. 
ph  /\  ps )
4 pm3.22 436 . . . . . 6  |-  ( ( -.  ph  /\  ps )  ->  ( ps  /\  -.  ph ) )
53, 4ax-mp 8 . . . . 5  |-  ( ps 
/\  -.  ph )
6 orc 374 . . . . 5  |-  ( ( ps  /\  -.  ph )  ->  ( ( ps 
/\  -.  ph )  \/  ( ph  /\  -.  ps ) ) )
75, 6ax-mp 8 . . . 4  |-  ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ps ) )
8 pm1.4 375 . . . 4  |-  ( ( ( ps  /\  -.  ph )  \/  ( ph  /\ 
-.  ps ) )  -> 
( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
97, 8ax-mp 8 . . 3  |-  ( (
ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) )
10 xor 861 . . . . 5  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )
11 bicom 191 . . . . . 6  |-  ( ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )  <->  ( (
( ph  /\  -.  ps )  \/  ( ps  /\ 
-.  ph ) )  <->  -.  ( ph 
<->  ps ) ) )
1211biimpi 186 . . . . 5  |-  ( ( -.  ( ph  <->  ps )  <->  ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
) )  ->  (
( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  -.  ( ph  <->  ps ) ) )
1310, 12ax-mp 8 . . . 4  |-  ( ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  <->  -.  ( ph  <->  ps ) )
1413biimpi 186 . . 3  |-  ( ( ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph )
)  ->  -.  ( ph 
<->  ps ) )
159, 14ax-mp 8 . 2  |-  -.  ( ph 
<->  ps )
16 df-xor 1296 . . . 4  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
17 bicom 191 . . . . 5  |-  ( ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
)  <->  ( -.  ( ph 
<->  ps )  <->  ( ph  \/_ 
ps ) ) )
1817biimpi 186 . . . 4  |-  ( ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
)  ->  ( -.  ( ph  <->  ps )  <->  ( ph  \/_ 
ps ) ) )
1916, 18ax-mp 8 . . 3  |-  ( -.  ( ph  <->  ps )  <->  (
ph  \/_  ps )
)
2019biimpi 186 . 2  |-  ( -.  ( ph  <->  ps )  ->  ( ph  \/_  ps ) )
2115, 20ax-mp 8 1  |-  ( ph  \/_ 
ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    \/_ wxo 1295
This theorem is referenced by:  aisfbistiaxb  27992
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-xor 1296
  Copyright terms: Public domain W3C validator