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Theorem mdandyv11 38419
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
mdandyv11.1  |-  ( ph  <-> F.  )
mdandyv11.2  |-  ( ps  <-> T.  )
mdandyv11.3  |-  ( ch  <-> T.  )
mdandyv11.4  |-  ( th  <-> T.  )
mdandyv11.5  |-  ( ta  <-> F.  )
mdandyv11.6  |-  ( et  <-> T.  )
Assertion
Ref Expression
mdandyv11  |-  ( ( ( ( ch  <->  ps )  /\  ( th  <->  ps )
)  /\  ( ta  <->  ph ) )  /\  ( et 
<->  ps ) )

Proof of Theorem mdandyv11
StepHypRef Expression
1 mdandyv11.3 . . . . 5  |-  ( ch  <-> T.  )
2 mdandyv11.2 . . . . 5  |-  ( ps  <-> T.  )
31, 2bothtbothsame 38357 . . . 4  |-  ( ch  <->  ps )
4 mdandyv11.4 . . . . 5  |-  ( th  <-> T.  )
54, 2bothtbothsame 38357 . . . 4  |-  ( th  <->  ps )
63, 5pm3.2i 456 . . 3  |-  ( ( ch  <->  ps )  /\  ( th 
<->  ps ) )
7 mdandyv11.5 . . . 4  |-  ( ta  <-> F.  )
8 mdandyv11.1 . . . 4  |-  ( ph  <-> F.  )
97, 8bothfbothsame 38358 . . 3  |-  ( ta  <->  ph )
106, 9pm3.2i 456 . 2  |-  ( ( ( ch  <->  ps )  /\  ( th  <->  ps )
)  /\  ( ta  <->  ph ) )
11 mdandyv11.6 . . 3  |-  ( et  <-> T.  )
1211, 2bothtbothsame 38357 . 2  |-  ( et  <->  ps )
1310, 12pm3.2i 456 1  |-  ( ( ( ( ch  <->  ps )  /\  ( 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