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Theorem mdandyvr13 27923
Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
Hypotheses
Ref Expression
mdandyvr13.1  |-  ( ph  <->  ze )
mdandyvr13.2  |-  ( ps  <->  si )
mdandyvr13.3  |-  ( ch  <->  ps )
mdandyvr13.4  |-  ( th  <->  ph )
mdandyvr13.5  |-  ( ta  <->  ps )
mdandyvr13.6  |-  ( et  <->  ps )
Assertion
Ref Expression
mdandyvr13  |-  ( ( ( ( ch  <->  si )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )  /\  ( et 
<-> 
si ) )

Proof of Theorem mdandyvr13
StepHypRef Expression
1 mdandyvr13.3 . . . . 5  |-  ( ch  <->  ps )
2 mdandyvr13.2 . . . . 5  |-  ( ps  <->  si )
31, 2bitri 240 . . . 4  |-  ( ch  <->  si )
4 mdandyvr13.4 . . . . 5  |-  ( th  <->  ph )
5 mdandyvr13.1 . . . . 5  |-  ( ph  <->  ze )
64, 5bitri 240 . . . 4  |-  ( th  <->  ze )
73, 6pm3.2i 441 . . 3  |-  ( ( ch  <->  si )  /\  ( th 
<->  ze ) )
8 mdandyvr13.5 . . . 4  |-  ( ta  <->  ps )
98, 2bitri 240 . . 3  |-  ( ta  <->  si )
107, 9pm3.2i 441 . 2  |-  ( ( ( ch  <->  si )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )
11 mdandyvr13.6 . . 3  |-  ( et  <->  ps )
1211, 2bitri 240 . 2  |-  ( et  <->  si )
1310, 12pm3.2i 441 1  |-  ( ( ( ( ch  <->  si )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )  /\  ( et 
<-> 
si ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358
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
  Copyright terms: Public domain W3C validator