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Theorem mdandyv4 27773
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
mdandyv4.1  |-  ( ph  <->  F.  )
mdandyv4.2  |-  ( ps  <->  T.  )
mdandyv4.3  |-  ( ch  <->  F.  )
mdandyv4.4  |-  ( th  <->  F.  )
mdandyv4.5  |-  ( ta  <->  T.  )
mdandyv4.6  |-  ( et  <->  F.  )
Assertion
Ref Expression
mdandyv4  |-  ( ( ( ( ch  <->  ph )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )

Proof of Theorem mdandyv4
StepHypRef Expression
1 mdandyv4.3 . . . . 5  |-  ( ch  <->  F.  )
2 mdandyv4.1 . . . . 5  |-  ( ph  <->  F.  )
31, 2bothfbothsame 27743 . . . 4  |-  ( ch  <->  ph )
4 mdandyv4.4 . . . . 5  |-  ( th  <->  F.  )
54, 2bothfbothsame 27743 . . . 4  |-  ( th  <->  ph )
63, 5pm3.2i 442 . . 3  |-  ( ( ch  <->  ph )  /\  ( th 
<-> 
ph ) )
7 mdandyv4.5 . . . 4  |-  ( ta  <->  T.  )
8 mdandyv4.2 . . . 4  |-  ( ps  <->  T.  )
97, 8bothtbothsame 27742 . . 3  |-  ( ta  <->  ps )
106, 9pm3.2i 442 . 2  |-  ( ( ( ch  <->  ph )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps ) )
11 mdandyv4.6 . . 3  |-  ( et  <->  F.  )
1211, 2bothfbothsame 27743 . 2  |-  ( et  <->  ph )
1310, 12pm3.2i 442 1  |-  ( ( ( ( ch  <->  ph )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1322    F. wfal 1323
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 178  df-an 361
  Copyright terms: Public domain W3C validator