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Theorem mdandyvr4 27904
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 242 . . . 4  |-  ( ch  <->  ze )
4 mdandyvr4.4 . . . . 5  |-  ( th  <->  ph )
54, 2bitri 242 . . . 4  |-  ( th  <->  ze )
63, 5pm3.2i 443 . . 3  |-  ( ( ch  <->  ze )  /\  ( th 
<->  ze ) )
7 mdandyvr4.5 . . . 4  |-  ( ta  <->  ps )
8 mdandyvr4.2 . . . 4  |-  ( ps  <->  si )
97, 8bitri 242 . . 3  |-  ( ta  <->  si )
106, 9pm3.2i 443 . 2  |-  ( ( ( ch  <->  ze )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )
11 mdandyvr4.6 . . 3  |-  ( et  <->  ph )
1211, 2bitri 242 . 2  |-  ( et  <->  ze )
1310, 12pm3.2i 443 1  |-  ( ( ( ( ch  <->  ze )  /\  ( th  <->  ze )
)  /\  ( ta  <->  si ) )  /\  ( et 
<->  ze ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360
This theorem is referenced by:  mdandyvr11  27911
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 179  df-an 362
  Copyright terms: Public domain W3C validator