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Theorem mdandyv12 28009
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
mdandyv12.1  |-  ( ph  <->  F.  )
mdandyv12.2  |-  ( ps  <->  T.  )
mdandyv12.3  |-  ( ch  <->  F.  )
mdandyv12.4  |-  ( th  <->  F.  )
mdandyv12.5  |-  ( ta  <->  T.  )
mdandyv12.6  |-  ( et  <->  T.  )
Assertion
Ref Expression
mdandyv12  |-  ( ( ( ( ch  <->  ph )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ps ) )

Proof of Theorem mdandyv12
StepHypRef Expression
1 mdandyv12.3 . . . . 5  |-  ( ch  <->  F.  )
2 mdandyv12.1 . . . . 5  |-  ( ph  <->  F.  )
31, 2bothfbothsame 27971 . . . 4  |-  ( ch  <->  ph )
4 mdandyv12.4 . . . . 5  |-  ( th  <->  F.  )
54, 2bothfbothsame 27971 . . . 4  |-  ( th  <->  ph )
63, 5pm3.2i 441 . . 3  |-  ( ( ch  <->  ph )  /\  ( th 
<-> 
ph ) )
7 mdandyv12.5 . . . 4  |-  ( ta  <->  T.  )
8 mdandyv12.2 . . . 4  |-  ( ps  <->  T.  )
97, 8bothtbothsame 27970 . . 3  |-  ( ta  <->  ps )
106, 9pm3.2i 441 . 2  |-  ( ( ( ch  <->  ph )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps ) )
11 mdandyv12.6 . . 3  |-  ( et  <->  T.  )
1211, 8bothtbothsame 27970 . 2  |-  ( et  <->  ps )
1310, 12pm3.2i 441 1  |-  ( ( ( ( ch  <->  ph )  /\  ( 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
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