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Theorem mdandyv5 38413
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
mdandyv5.1  |-  ( ph  <-> F.  )
mdandyv5.2  |-  ( ps  <-> T.  )
mdandyv5.3  |-  ( ch  <-> T.  )
mdandyv5.4  |-  ( th  <-> F.  )
mdandyv5.5  |-  ( ta  <-> T.  )
mdandyv5.6  |-  ( et  <-> F.  )
Assertion
Ref Expression
mdandyv5  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps )
)  /\  ( et  <->  ph ) )

Proof of Theorem mdandyv5
StepHypRef Expression
1 mdandyv5.3 . . . . 5  |-  ( ch  <-> T.  )
2 mdandyv5.2 . . . . 5  |-  ( ps  <-> T.  )
31, 2bothtbothsame 38357 . . . 4  |-  ( ch  <->  ps )
4 mdandyv5.4 . . . . 5  |-  ( th  <-> F.  )
5 mdandyv5.1 . . . . 5  |-  ( ph  <-> F.  )
64, 5bothfbothsame 38358 . . . 4  |-  ( th  <->  ph )
73, 6pm3.2i 456 . . 3  |-  ( ( ch  <->  ps )  /\  ( th 
<-> 
ph ) )
8 mdandyv5.5 . . . 4  |-  ( ta  <-> T.  )
98, 2bothtbothsame 38357 . . 3  |-  ( ta  <->  ps )
107, 9pm3.2i 456 . 2  |-  ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps )
)
11 mdandyv5.6 . . 3  |-  ( et  <-> F.  )
1211, 5bothfbothsame 38358 . 2  |-  ( et  <->  ph )
1310, 12pm3.2i 456 1  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ps )
)  /\  ( et  <->  ph ) )
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