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Theorem mdandyvrx0 28029
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
mdandyvrx0.1  |-  ( ph  \/_ 
ze )
mdandyvrx0.2  |-  ( ps 
\/_  si )
mdandyvrx0.3  |-  ( ch  <->  ph )
mdandyvrx0.4  |-  ( th  <->  ph )
mdandyvrx0.5  |-  ( ta  <->  ph )
mdandyvrx0.6  |-  ( et  <->  ph )
Assertion
Ref Expression
mdandyvrx0  |-  ( ( ( ( ch  \/_  ze )  /\  ( th 
\/_  ze ) )  /\  ( ta  \/_  ze )
)  /\  ( et  \/_  ze ) )

Proof of Theorem mdandyvrx0
StepHypRef Expression
1 mdandyvrx0.1 . . . . 5  |-  ( ph  \/_ 
ze )
2 mdandyvrx0.3 . . . . 5  |-  ( ch  <->  ph )
31, 2axorbciffatcxorb 27976 . . . 4  |-  ( ch 
\/_  ze )
4 mdandyvrx0.4 . . . . 5  |-  ( th  <->  ph )
51, 4axorbciffatcxorb 27976 . . . 4  |-  ( th 
\/_  ze )
63, 5pm3.2i 441 . . 3  |-  ( ( ch  \/_  ze )  /\  ( th  \/_  ze ) )
7 mdandyvrx0.5 . . . 4  |-  ( ta  <->  ph )
81, 7axorbciffatcxorb 27976 . . 3  |-  ( ta 
\/_  ze )
96, 8pm3.2i 441 . 2  |-  ( ( ( ch  \/_  ze )  /\  ( th  \/_  ze ) )  /\  ( ta  \/_  ze ) )
10 mdandyvrx0.6 . . 3  |-  ( et  <->  ph )
111, 10axorbciffatcxorb 27976 . 2  |-  ( et 
\/_  ze )
129, 11pm3.2i 441 1  |-  ( ( ( ( ch  \/_  ze )  /\  ( th 
\/_  ze ) )  /\  ( ta  \/_  ze )
)  /\  ( et  \/_  ze ) )
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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