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Theorem mdandyvrx4 27600
Description: Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
Hypotheses
Ref Expression
mdandyvrx4.1  |-  ( ph  \/_ 
ze )
mdandyvrx4.2  |-  ( ps 
\/_  si )
mdandyvrx4.3  |-  ( ch  <->  ph )
mdandyvrx4.4  |-  ( th  <->  ph )
mdandyvrx4.5  |-  ( ta  <->  ps )
mdandyvrx4.6  |-  ( et  <->  ph )
Assertion
Ref Expression
mdandyvrx4  |-  ( ( ( ( ch  \/_  ze )  /\  ( th 
\/_  ze ) )  /\  ( ta  \/_  si )
)  /\  ( et  \/_  ze ) )

Proof of Theorem mdandyvrx4
StepHypRef Expression
1 mdandyvrx4.1 . . . . 5  |-  ( ph  \/_ 
ze )
2 mdandyvrx4.3 . . . . 5  |-  ( ch  <->  ph )
31, 2axorbciffatcxorb 27543 . . . 4  |-  ( ch 
\/_  ze )
4 mdandyvrx4.4 . . . . 5  |-  ( th  <->  ph )
51, 4axorbciffatcxorb 27543 . . . 4  |-  ( th 
\/_  ze )
63, 5pm3.2i 442 . . 3  |-  ( ( ch  \/_  ze )  /\  ( th  \/_  ze ) )
7 mdandyvrx4.2 . . . 4  |-  ( ps 
\/_  si )
8 mdandyvrx4.5 . . . 4  |-  ( ta  <->  ps )
97, 8axorbciffatcxorb 27543 . . 3  |-  ( ta 
\/_  si )
106, 9pm3.2i 442 . 2  |-  ( ( ( ch  \/_  ze )  /\  ( th  \/_  ze ) )  /\  ( ta  \/_  si ) )
11 mdandyvrx4.6 . . 3  |-  ( et  <->  ph )
121, 11axorbciffatcxorb 27543 . 2  |-  ( et 
\/_  ze )
1310, 12pm3.2i 442 1  |-  ( ( ( ( ch  \/_  ze )  /\  ( th 
\/_  ze ) )  /\  ( ta  \/_  si )
)  /\  ( et  \/_  ze ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    \/_ wxo 1310
This theorem is referenced by:  mdandyvrx11  27607
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  df-xor 1311
  Copyright terms: Public domain W3C validator