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Theorem aisbnaxb 27982
Description: Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)
Hypothesis
Ref Expression
aisbnaxb.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
aisbnaxb  |-  -.  ( ph  \/_  ps )

Proof of Theorem aisbnaxb
StepHypRef Expression
1 aisbnaxb.1 . . 3  |-  ( ph  <->  ps )
2 notnot 282 . . . 4  |-  ( (
ph 
<->  ps )  <->  -.  -.  ( ph 
<->  ps ) )
32biimpi 186 . . 3  |-  ( (
ph 
<->  ps )  ->  -.  -.  ( ph  <->  ps )
)
41, 3ax-mp 8 . 2  |-  -.  -.  ( ph  <->  ps )
5 df-xor 1296 . . . . 5  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
6 notbi 286 . . . . . 6  |-  ( ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
)  <->  ( -.  ( ph  \/_  ps )  <->  -.  -.  ( ph 
<->  ps ) ) )
76biimpi 186 . . . . 5  |-  ( ( ( ph  \/_  ps ) 
<->  -.  ( ph  <->  ps )
)  ->  ( -.  ( ph  \/_  ps )  <->  -. 
-.  ( ph  <->  ps )
) )
85, 7ax-mp 8 . . . 4  |-  ( -.  ( ph  \/_  ps ) 
<->  -.  -.  ( ph  <->  ps ) )
9 bicom 191 . . . . 5  |-  ( ( -.  ( ph  \/_  ps ) 
<->  -.  -.  ( ph  <->  ps ) )  <->  ( -.  -.  ( ph  <->  ps )  <->  -.  ( ph  \/_  ps ) ) )
109biimpi 186 . . . 4  |-  ( ( -.  ( ph  \/_  ps ) 
<->  -.  -.  ( ph  <->  ps ) )  ->  ( -.  -.  ( ph  <->  ps )  <->  -.  ( ph  \/_  ps ) ) )
118, 10ax-mp 8 . . 3  |-  ( -. 
-.  ( ph  <->  ps )  <->  -.  ( ph  \/_  ps ) )
1211biimpi 186 . 2  |-  ( -. 
-.  ( ph  <->  ps )  ->  -.  ( ph  \/_  ps ) )
134, 12ax-mp 8 1  |-  -.  ( ph  \/_  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/_ wxo 1295
This theorem is referenced by:  dandysum2p2e4  28046
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-xor 1296
  Copyright terms: Public domain W3C validator