Users' Mathboxes Mathbox for Jarvin Udandy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mdandyvr4 Unicode version

Theorem mdandyvr4 27914
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
mdandyvr4.1  |-  ( ph  <->  ze )
mdandyvr4.2  |-  ( ps  <->  si )
mdandyvr4.3  |-  ( ch  <->  ph )
mdandyvr4.4  |-  ( th  <->  ph )
mdandyvr4.5  |-  ( ta  <->  ps )
mdandyvr4.6  |-  ( et  <->  ph )
Assertion
Ref Expression
mdandyvr4  |-  ( ( ( ( ch  <->  ze )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )  /\  ( et 
<->  ze ) )

Proof of Theorem mdandyvr4
StepHypRef Expression
1 mdandyvr4.3 . . . . 5  |-  ( ch  <->  ph )
2 mdandyvr4.1 . . . . 5  |-  ( ph  <->  ze )
31, 2bitri 240 . . . 4  |-  ( ch  <->  ze )
4 mdandyvr4.4 . . . . 5  |-  ( th  <->  ph )
54, 2bitri 240 . . . 4  |-  ( th  <->  ze )
63, 5pm3.2i 441 . . 3  |-  ( ( ch  <->  ze )  /\  ( th 
<->  ze ) )
7 mdandyvr4.5 . . . 4  |-  ( ta  <->  ps )
8 mdandyvr4.2 . . . 4  |-  ( ps  <->  si )
97, 8bitri 240 . . 3  |-  ( ta  <->  si )
106, 9pm3.2i 441 . 2  |-  ( ( ( ch  <->  ze )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )
11 mdandyvr4.6 . . 3  |-  ( et  <->  ph )
1211, 2bitri 240 . 2  |-  ( et  <->  ze )
1310, 12pm3.2i 441 1  |-  ( ( ( ( ch  <->  ze )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )  /\  ( et 
<->  ze ) )
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