Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mdandyv9 Structured version   Unicode version

Theorem mdandyv9 27879
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
mdandyv9.1  |-  ( ph  <->  F.  )
mdandyv9.2  |-  ( ps  <->  T.  )
mdandyv9.3  |-  ( ch  <->  T.  )
mdandyv9.4  |-  ( th  <->  F.  )
mdandyv9.5  |-  ( ta  <->  F.  )
mdandyv9.6  |-  ( et  <->  T.  )
Assertion
Ref Expression
mdandyv9  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps )
)

Proof of Theorem mdandyv9
StepHypRef Expression
1 mdandyv9.3 . . . . 5  |-  ( ch  <->  T.  )
2 mdandyv9.2 . . . . 5  |-  ( ps  <->  T.  )
31, 2bothtbothsame 27843 . . . 4  |-  ( ch  <->  ps )
4 mdandyv9.4 . . . . 5  |-  ( th  <->  F.  )
5 mdandyv9.1 . . . . 5  |-  ( ph  <->  F.  )
64, 5bothfbothsame 27844 . . . 4  |-  ( th  <->  ph )
73, 6pm3.2i 442 . . 3  |-  ( ( ch  <->  ps )  /\  ( th 
<-> 
ph ) )
8 mdandyv9.5 . . . 4  |-  ( ta  <->  F.  )
98, 5bothfbothsame 27844 . . 3  |-  ( ta  <->  ph )
107, 9pm3.2i 442 . 2  |-  ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ph ) )
11 mdandyv9.6 . . 3  |-  ( et  <->  T.  )
1211, 2bothtbothsame 27843 . 2  |-  ( et  <->  ps )
1310, 12pm3.2i 442 1  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1325    F. wfal 1326
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