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Theorem aiffnbandciffatnotciffb 27872
Description: Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)
Hypotheses
Ref Expression
aiffnbandciffatnotciffb.1  |-  ( ph  <->  -. 
ps )
aiffnbandciffatnotciffb.2  |-  ( ch  <->  ph )
Assertion
Ref Expression
aiffnbandciffatnotciffb  |-  -.  ( ch 
<->  ps )

Proof of Theorem aiffnbandciffatnotciffb
StepHypRef Expression
1 aiffnbandciffatnotciffb.2 . . 3  |-  ( ch  <->  ph )
2 aiffnbandciffatnotciffb.1 . . 3  |-  ( ph  <->  -. 
ps )
31, 2bitri 240 . 2  |-  ( ch  <->  -. 
ps )
4 xor3 346 . . . 4  |-  ( -.  ( ch  <->  ps )  <->  ( ch  <->  -.  ps )
)
5 bicom 191 . . . . 5  |-  ( ( -.  ( ch  <->  ps )  <->  ( ch  <->  -.  ps )
)  <->  ( ( ch  <->  -. 
ps )  <->  -.  ( ch 
<->  ps ) ) )
65biimpi 186 . . . 4  |-  ( ( -.  ( ch  <->  ps )  <->  ( ch  <->  -.  ps )
)  ->  ( ( ch 
<->  -.  ps )  <->  -.  ( ch 
<->  ps ) ) )
74, 6ax-mp 8 . . 3  |-  ( ( ch  <->  -.  ps )  <->  -.  ( ch  <->  ps )
)
87biimpi 186 . 2  |-  ( ( ch  <->  -.  ps )  ->  -.  ( ch  <->  ps )
)
93, 8ax-mp 8 1  |-  -.  ( ch 
<->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176
This theorem is referenced by:  axorbciffatcxorb  27873
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
  Copyright terms: Public domain W3C validator