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Theorem mdandyv13 27907
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
Hypotheses
Ref Expression
mdandyv13.1  |-  ( ph  <->  F.  )
mdandyv13.2  |-  ( ps  <->  T.  )
mdandyv13.3  |-  ( ch  <->  T.  )
mdandyv13.4  |-  ( th  <->  F.  )
mdandyv13.5  |-  ( ta  <->  T.  )
mdandyv13.6  |-  ( et  <->  T.  )
Assertion
Ref Expression
mdandyv13  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps )
)  /\  ( et  <->  ps ) )

Proof of Theorem mdandyv13
StepHypRef Expression
1 mdandyv13.3 . . . . 5  |-  ( ch  <->  T.  )
2 mdandyv13.2 . . . . 5  |-  ( ps  <->  T.  )
31, 2bothtbothsame 27867 . . . 4  |-  ( ch  <->  ps )
4 mdandyv13.4 . . . . 5  |-  ( th  <->  F.  )
5 mdandyv13.1 . . . . 5  |-  ( ph  <->  F.  )
64, 5bothfbothsame 27868 . . . 4  |-  ( th  <->  ph )
73, 6pm3.2i 441 . . 3  |-  ( ( ch  <->  ps )  /\  ( th 
<-> 
ph ) )
8 mdandyv13.5 . . . 4  |-  ( ta  <->  T.  )
98, 2bothtbothsame 27867 . . 3  |-  ( ta  <->  ps )
107, 9pm3.2i 441 . 2  |-  ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps )
)
11 mdandyv13.6 . . 3  |-  ( et  <->  T.  )
1211, 2bothtbothsame 27867 . 2  |-  ( et  <->  ps )
1310, 12pm3.2i 441 1  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps )
)  /\  ( et  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    F. wfal 1308
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-an 360  df-tru 1310  df-fal 1311
  Copyright terms: Public domain W3C validator