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Theorem mdandyv10 38418
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
mdandyv10.1  |-  ( ph  <-> F.  )
mdandyv10.2  |-  ( ps  <-> T.  )
mdandyv10.3  |-  ( ch  <-> F.  )
mdandyv10.4  |-  ( th  <-> T.  )
mdandyv10.5  |-  ( ta  <-> F.  )
mdandyv10.6  |-  ( et  <-> T.  )
Assertion
Ref Expression
mdandyv10  |-  ( ( ( ( ch  <->  ph )  /\  ( th  <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps ) )

Proof of Theorem mdandyv10
StepHypRef Expression
1 mdandyv10.3 . . . . 5  |-  ( ch  <-> F.  )
2 mdandyv10.1 . . . . 5  |-  ( ph  <-> F.  )
31, 2bothfbothsame 38358 . . . 4  |-  ( ch  <->  ph )
4 mdandyv10.4 . . . . 5  |-  ( th  <-> T.  )
5 mdandyv10.2 . . . . 5  |-  ( ps  <-> T.  )
64, 5bothtbothsame 38357 . . . 4  |-  ( th  <->  ps )
73, 6pm3.2i 456 . . 3  |-  ( ( ch  <->  ph )  /\  ( th 
<->  ps ) )
8 mdandyv10.5 . . . 4  |-  ( ta  <-> F.  )
98, 2bothfbothsame 38358 . . 3  |-  ( ta  <->  ph )
107, 9pm3.2i 456 . 2  |-  ( ( ( ch  <->  ph )  /\  ( th  <->  ps ) )  /\  ( ta  <->  ph ) )
11 mdandyv10.6 . . 3  |-  ( et  <-> T.  )
1211, 5bothtbothsame 38357 . 2  |-  ( et  <->  ps )
1310, 12pm3.2i 456 1  |-  ( ( ( ( ch  <->  ph )  /\  ( th  <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   T. wtru 1438   F. wfal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-an 372
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator