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Theorem mdandyvrx12 28041
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
mdandyvrx12.1  |-  ( ph  \/_ 
ze )
mdandyvrx12.2  |-  ( ps 
\/_  si )
mdandyvrx12.3  |-  ( ch  <->  ph )
mdandyvrx12.4  |-  ( th  <->  ph )
mdandyvrx12.5  |-  ( ta  <->  ps )
mdandyvrx12.6  |-  ( et  <->  ps )
Assertion
Ref Expression
mdandyvrx12  |-  ( ( ( ( ch  \/_  ze )  /\  ( th 
\/_  ze ) )  /\  ( ta  \/_  si )
)  /\  ( et  \/_  si ) )

Proof of Theorem mdandyvrx12
StepHypRef Expression
1 mdandyvrx12.1 . . . . 5  |-  ( ph  \/_ 
ze )
2 mdandyvrx12.3 . . . . 5  |-  ( ch  <->  ph )
31, 2axorbciffatcxorb 27976 . . . 4  |-  ( ch 
\/_  ze )
4 mdandyvrx12.4 . . . . 5  |-  ( th  <->  ph )
51, 4axorbciffatcxorb 27976 . . . 4  |-  ( th 
\/_  ze )
63, 5pm3.2i 441 . . 3  |-  ( ( ch  \/_  ze )  /\  ( th  \/_  ze ) )
7 mdandyvrx12.2 . . . 4  |-  ( ps 
\/_  si )
8 mdandyvrx12.5 . . . 4  |-  ( ta  <->  ps )
97, 8axorbciffatcxorb 27976 . . 3  |-  ( ta 
\/_  si )
106, 9pm3.2i 441 . 2  |-  ( ( ( ch  \/_  ze )  /\  ( th  \/_  ze ) )  /\  ( ta  \/_  si ) )
11 mdandyvrx12.6 . . 3  |-  ( et  <->  ps )
127, 11axorbciffatcxorb 27976 . 2  |-  ( et 
\/_  si )
1310, 12pm3.2i 441 1  |-  ( ( ( ( ch  \/_  ze )  /\  ( th 
\/_  ze ) )  /\  ( ta  \/_  si )
)  /\  ( et  \/_  si ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    \/_ wxo 1295
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  df-xor 1296
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