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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabcdta 27301 Given (((a and b) and c) and d), there exists a proof for a (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |-  ph
 
Theoremabcdtb 27302 Given (((a and b) and c) and d), there exists a proof for b (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |- 
 ps
 
Theoremabcdtc 27303 Given (((a and b) and c) and d), there exists a proof for c (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |- 
 ch
 
Theoremabcdtd 27304 Given (((a and b) and c) and d), there exists a proof for d (Contributed by Jarvin Udandy, 3-Sep-2016.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  /\  th )   =>    |- 
 th
 
Theoremmdandyv0 27305 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv1 27306 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv2 27307 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv3 27308 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  F.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv4 27309 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  T.  )   &    |-  ( et 
 <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv5 27310 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  T.  )   &    |-  ( et 
 <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv6 27311 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  T.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv7 27312 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  T.  )   &    |-  ( et  <->  F.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ph ) )
 
Theoremmdandyv8 27313 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  T.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv9 27314 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  T.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv10 27315 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  F.  )   &    |-  ( et  <->  T.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv11 27316 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  F.  )   &    |-  ( et  <->  T.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ph ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv12 27317 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  T.  )   &    |-  ( et 
 <->  T.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv13 27318 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  T.  )   &    |-  ( et 
 <->  T.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <-> 
 ph ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv14 27319 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  F.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  T.  )   &    |-  ( et  <->  T.  )   =>    |-  ( ( ( ( ch  <->  ph )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyv15 27320 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)
 |-  ( ph 
 <->  F.  )   &    |-  ( ps  <->  T.  )   &    |-  ( ch 
 <->  T.  )   &    |-  ( th  <->  T.  )   &    |-  ( ta 
 <->  T.  )   &    |-  ( et  <->  T.  )   =>    |-  ( ( ( ( ch  <->  ps )  /\  ( th 
 <->  ps ) )  /\  ( ta  <->  ps ) )  /\  ( et  <->  ps ) )
 
Theoremmdandyvr0 27321 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr1 27322 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr2 27323 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr3 27324 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr4 27325 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr5 27326 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr6 27327 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr7 27328 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  ze ) )
 
Theoremmdandyvr8 27329 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr9 27330 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr10 27331 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr11 27332 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  ze ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr12 27333 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr13 27334 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr14 27335 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr15 27336 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.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvrx0 27337 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx1 27338 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx2 27339 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx3 27340 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx4 27341 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx5 27342 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx6 27343 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ ze ) )
 
Theoremmdandyvrx7 27344 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ ze ) )
 
Theoremmdandyvrx8 27345 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx9 27346 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx10 27347 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx11 27348 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx12 27349 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx13 27350 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx14 27351 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ si ) )
 
Theoremmdandyvrx15 27352 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.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ si ) )
 
TheoremH15NH16TH15IH16 27353 Given 15 hypotheses and a 16th hypothesis, there exists a proof the 15 imply the 16th. (Contributed by Jarvin Udandy, 8-Sep-2016.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ta   &    |-  et   &    |-  ze   &    |-  si   &    |-  rh   &    |-  mu   &    |-  la   &    |-  ka   &    |- jph   &    |- jps   &    |- jch   &    |- jth   =>    |-  (
 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  /\ jph )  /\ jps
 )  /\ jch ) 
 -> jth )
 
Theoremdandysum2p2e4 27354

CONTRADICTION PROVED AT 1 + 1 = 2 .

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit. (Contributed by Jarvin Udandy, 6-Sep-2016.)

 |-  ( ph 
 <->  ( th  /\  ta ) )   &    |-  ( ps  <->  ( et  /\  ze ) )   &    |-  ( ch  <->  ( si  /\  rh ) )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  T.  )   &    |-  ( ze 
 <->  T.  )   &    |-  ( si  <->  F.  )   &    |-  ( rh  <->  F.  )   &    |-  ( mu  <->  F.  )   &    |-  ( la  <->  F.  )   &    |-  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) )   &    |-  (jph  <->  (
 ( et \/_ ze )  \/  ph ) )   &    |-  (jps  <->  ( ( si \/_ rh )  \/  ps ) )   &    |-  (jch  <->  ( ( mu
 \/_ la )  \/  ch ) )   =>    |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
 ph 
 <->  ( th  /\  ta ) )  /\  ( ps  <->  ( et  /\  ze )
 ) )  /\  ( ch 
 <->  ( si  /\  rh ) ) )  /\  ( th  <->  F.  ) )  /\  ( ta  <->  F.  ) )  /\  ( et  <->  T.  ) )  /\  ( ze  <->  T.  ) )  /\  ( si  <->  F.  ) )  /\  ( rh  <->  F.  ) )  /\  ( mu  <->  F.  ) )  /\  ( la  <->  F.  ) )  /\  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) ) )  /\  (jph  <->  ( ( et \/_ ze )  \/  ph ) ) ) 
 /\  (jps  <->  (
 ( si \/_ rh )  \/  ps ) ) ) 
 /\  (jch  <->  (
 ( mu \/_ la )  \/  ch ) ) ) 
 ->  ( ( ( ( ka  <->  F.  )  /\  (jph  <->  F.  ) )  /\  (jps  <->  T.  ) )  /\  (jch  <->  F.  ) ) )
 
Theoremmdandysum2p2e4 27355 CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half-adders and full adders in propositional calculus

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

 |-  (jth  <->  F.  )   &    |-  (jta  <->  T.  )   &    |-  ( ph  <->  ( th  /\  ta ) )   &    |-  ( ps  <->  ( et  /\  ze ) )   &    |-  ( ch  <->  ( si  /\  rh ) )   &    |-  ( th  <-> jth )   &    |-  ( ta 
 <-> jth
 )   &    |-  ( et  <-> jta )   &    |-  ( ze 
 <-> jta
 )   &    |-  ( si  <-> jth )   &    |-  ( rh 
 <-> jth
 )   &    |-  ( mu  <-> jth )   &    |-  ( la 
 <-> jth
 )   &    |-  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) )   &    |-  (jph  <->  (
 ( et \/_ ze )  \/  ph ) )   &    |-  (jps  <->  ( ( si \/_ rh )  \/  ps ) )   &    |-  (jch  <->  ( ( mu
 \/_ la )  \/  ch ) )   =>    |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
 ph 
 <->  ( th  /\  ta ) )  /\  ( ps  <->  ( et  /\  ze )
 ) )  /\  ( ch 
 <->  ( si  /\  rh ) ) )  /\  ( th  <->  F.  ) )  /\  ( ta  <->  F.  ) )  /\  ( et  <->  T.  ) )  /\  ( ze  <->  T.  ) )  /\  ( si  <->  F.  ) )  /\  ( rh  <->  F.  ) )  /\  ( mu  <->  F.  ) )  /\  ( la  <->  F.  ) )  /\  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) ) )  /\  (jph  <->  ( ( et \/_ ze )  \/  ph ) ) ) 
 /\  (jps  <->  (
 ( si \/_ rh )  \/  ps ) ) ) 
 /\  (jch  <->  (
 ( mu \/_ la )  \/  ch ) ) ) 
 ->  ( ( ( ( ka  <->  F.  )  /\  (jph  <->  F.  ) )  /\  (jps  <->  T.  ) )  /\  (jch  <->  F.  ) ) )
 
18.23  Mathbox for Alexander van der Vekens
 
18.23.1  Double restricted existential uniqueness
 
18.23.1.1  Restricted quantification (extension)
 
Theoremr19.32 27356 Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 2686. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  F/ x ph   =>    |-  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) )
 
Theoremrexsb 27357* An equivalent expression for restricted existence, analogous to exsb 2069. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x ( x  =  y  ->  ph ) )
 
Theoremrexrsb 27358* An equivalent expression for restricted existence, analogous to exsb 2069. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  ( x  =  y  ->  ph ) )
 
Theorem2rexsb 27359* An equivalent expression for double restricted existence, analogous to rexsb 27357. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  A. x A. y ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theorem2rexrsb 27360* An equivalent expression for double restricted existence, analogous to 2exsb 2071. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremcbvral2 27361* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 2772. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  F/ z ph   &    |-  F/ x ch   &    |-  F/ w ch   &    |-  F/ y ps   &    |-  ( x  =  z  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
 
Theoremcbvrex2 27362* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 2773. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  F/ z ph   &    |-  F/ x ch   &    |-  F/ w ch   &    |-  F/ y ps   &    |-  ( x  =  z  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  ps )
 
Theorem2ralbiim 27363 Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1599 and ralbiim 2680. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( A. x  e.  A  A. y  e.  B  (
 ph 
 <->  ps )  <->  ( A. x  e.  A  A. y  e.  B  ( ph  ->  ps )  /\  A. x  e.  A  A. y  e.  B  ( ps  ->  ph ) ) )
 
18.23.1.2  The empty set (extension)
 
Theoremraaan2 27364* Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3561. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  (
 ( A  =  (/)  <->  B  =  (/) )  ->  ( A. x  e.  A  A. y  e.  B  (
 ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  A. y  e.  B  ps ) ) )
 
18.23.1.3  Restricted uniqueness and "at most one" quantification
 
Theoremrmoimi 27365 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x  e.  A ps  ->  E* x  e.  A ph )
 
Theorem2reu5a 27366 Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E. x  e.  A  ( E. y  e.  B  ph 
 /\  E* y  e.  B ph )  /\  E* x  e.  A ( E. y  e.  B  ph  /\  E* y  e.  B ph ) ) )
 
Theoremreuimrmo 27367 Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2192. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( E! x  e.  A  ps  ->  E* x  e.  A ph ) )
 
Theoremrmoanim 27368* Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2199. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  F/ x ph   =>    |-  ( E* x  e.  A ( ph  /\  ps ) 
 <->  ( ph  ->  E* x  e.  A ps ) )
 
Theoremreuan 27369* Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2200. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  F/ x ph   =>    |-  ( E! x  e.  A  ( ph  /\  ps ) 
 <->  ( ph  /\  E! x  e.  A  ps ) )
 
18.23.1.4  Analogs to Existential uniqueness (double quantification)
 
Theorem2reurex 27370* Double restricted quantification with existential uniqueness, analogous to 2euex 2215. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
 |-  ( E! x  e.  A  E. y  e.  B  ph 
 ->  E. y  e.  B  E! x  e.  A  ph )
 
Theorem2reurmo 27371* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2216. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
 |-  ( E! x  e.  A  E* y  e.  B ph 
 ->  E* x  e.  A E! y  e.  B  ph )
 
Theorem2reu2rex 27372* Double restricted existential uniqueness, analogous to 2eu2ex 2217. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ph 
 ->  E. x  e.  A  E. y  e.  B  ph )
 
Theorem2rmoswap 27373* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2218. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( A. x  e.  A  E* y  e.  B ph 
 ->  ( E* x  e.  A E. y  e.  B  ph  ->  E* y  e.  B E. x  e.  A  ph ) )
 
Theorem2rexreu 27374* Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2220. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  ->  E! x  e.  A  E! y  e.  B  ph )
 
Theorem2reu1 27375* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2223. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( A. x  e.  A  E* y  e.  B ph 
 ->  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph ) ) )
 
Theorem2reu2 27376* Double restricted existential uniqueness, analogous to 2eu2 2224. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  ( E! y  e.  B  E. x  e.  A  ph 
 ->  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E. y  e.  B  ph ) )
 
Theorem2reu3 27377* Double restricted existential uniqueness, analogous to 2eu3 2225. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  ( A. x  e.  A  A. y  e.  B  ( E* x  e.  A ph 
 \/  E* y  e.  B ph )  ->  ( ( E! x  e.  A  E! y  e.  B  ph 
 /\  E! y  e.  B  E! x  e.  A  ph )  <->  ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph ) ) )
 
Theorem2reu4a 27378* Definition of double restricted existential uniqueness ("exactly one  x and exactly one  y"), analogous to 2eu4 2226 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 27379). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  (
 ( A  =/=  (/)  /\  B  =/= 
 (/) )  ->  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) ) )
 
Theorem2reu4 27379* Definition of double restricted existential uniqueness ("exactly one  x and exactly one  y"), analogous to 2eu4 2226. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) )
 
Theorem2reu7 27380* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2229. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  E! x  e.  A  E! y  e.  B  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ph ) )
 
Theorem2reu8 27381* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2230. Curiously, we can put  E! on either of the internal conjuncts but not both. We can also commute  E! x  e.  A E! y  e.  B using 2reu7 27380. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ph )  <->  E! x  e.  A  E! y  e.  B  ( E! x  e.  A  ph 
 /\  E. y  e.  B  ph ) )
 
18.23.2  Alternative definitions of function's and operation's values

The current definition of the value 
( F `  A
) of a function  F for an argument  A ( see df-fv 5228) assures that this values is always a set, see fex 5710. This is because this definition can be applied to any classes  F and  A, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 5513 and fvprc 5482).

Although is is very convenient for many theorems on functions and their proofs, there are some cases in which from  ( F `  A
)  =  (/) alone it cannot be decided/derived if  ( F `  A ) is meaningful ( F is actually a function which is defined for  A and really has the function value  (/)) or not. Therefore, additional assumptions are required, such as  (/)  e/  ran  F,  (/)  e.  ran  F or 
Fun  F  /\  A  e. 
dom  F (see, for example, ndmfvrcl 5514).

To avoid such an ambiguity, an alternative definition  ( F''' A ) ( see df-afv 27386) would be possible which evaluates to the universal class ( ( F''' A )  =  _V) if it is not meaningful (see afvnfundmuv 27413, ndmafv 27414, afvprc 27418 and nfunsnafv 27416), and which corresponds to the current definition ( ( F `  A )  =  ( F''' A )) if it is (see afvfundmfveq 27412). That means  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) (see afvpcfv0 27420).

In the theory of partial functions, it is a common case that  F is not defined at  A, which also would result in  ( F''' A )  =  _V. In this context we say  ( F''' A ) "is not defined" instead of "is not meaningful".

With this definition the following intuitive equivalence holds:  ( F''' A )  e.  _V <-> " ( F''' A ) is meaningful/defined"

An interesting question would be if 
( F `  A
) could be replaced by  ( F''' A ) in most of the theorems based on function's values. If we look at the (currently 15) proofs using the definition df-fv 5228 of 
( F `  A
), we see that analogons for the following 6 theorems can be proven using the alternative definition: fveq1 5484-> afveq1 27408, fveq2 5485-> afveq2 27409, nffv 5492-> nfafv 27410, csbfv12g 5495-> csbafv12g , fvres 5502-> afvres 27445, fvco2 5555-> afvco2 27448. From these, only afvco2 27448 is a little bit tricky to be proven. 3 theorems proved by directly using df-fv 5228 are deprecated (usage discouraged) or within mathboxes: csbfv12gALT 5496, repfuntw 24571, csbfv12gALTVD 27978. 2 additional theorems proved by directly using df-fv 5228 can also be neclected: dffv3 6280 (used only in dffv4 23872 which is contained in a mathbox) and avril1 20829 (not used ;-)). However, the remaining 4 theorems proved by directly using df-fv 5228 are used more or less often: fv2 5481 (see below), fvprc 5482 (used in 127 proofs), fvex 5499 (used in about 1750 proofs), shftval 11564 (used in 7 proofs). These 4 theorems are not valid in general for the alternative definition, so each proof using them must be examinated if these theorems can be replaced by other theorems, maybe adding additional assertions. For fv2 5481, for example, there are 3 theorems being proved by using fv2 5481: elfv 5483, fv4 6281 and ovtpos 6210. While ovtpos 6210 is used in 14 proofs, elfv 5483 is used (indirectly via tz6.12-2 5507, tz6.12i 5509, and fvbr0 5510) for fvrn0 5511 (used in 18 proofs) , dcomex 8068 (used in 4 proofs) and ndmfv 5513 (used in 86 proofs). fv3 5501, which is also proven using elfv 5483, is only used by tz6.12-1 5504 which could be replaced by tz6.12-afv 27446! Finally, regarding elfv 5483, the proof of nfunsn 5519 uses tz6.12-2 5507, but nfunsn 5519 is only used for dffv2 5553 which itself is not used in any proof. The remaining theorem fv4 6281 whose proof uses fv2 5481 is (indirectly) used for adjbdln 22658 (used in 7 proofs ), isum 12187 (used in 7 proofs ), sum0 12189 (used in 42 proofs ), fsumser 12198 (used in 40 proofs ) and logtayl 20002 (used in 4 proofs).

As a result of this analysis we can say that the current definition of a function's value is crucial for Methamath and cannot be exchanged with easily an alternative definition.

With the same arguments, an alternatvie definition of operation's values (( A O B)) could be meaningful to avoid ambiguites, see df-aov 27387.

For additional discussions/material see https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4.

 
Syntaxwdfat 27382 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function)  F is defined at (the argument)  A). In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for $ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
 wff  F defAt  A
 
Syntaxcafv 27383 Extend the definition of a class to include the value of a function. (Read: The value of  F at  A, or " F of  A."). In a previous version, the symbol " ' " was used. However, since the similarity with the symbol 
` used for the current definition of a function's value (see df-fv 5228), which, by the way, was intended to visualize that in many cases  ` and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 27406, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5228 and df-ima 4700. And not three backticks ( three times  ` ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
 class  ( F''' A )
 
Syntaxcaov 27384 Extend class notation to include the value of an operation  F (such as  +) for two arguments  A and  B. Note that the syntax is simply three class symbols in a row surrounded by special parentheses (exclamation mark with underscore) in contrast to the current definition, see df-ov 5822.
 class (( A F B))
 
Definitiondf-dfat 27385 Definition of the predicate that determines if some class  F is defined as function for an argument  A or, in other words, if the function value for some class  F for an argument  A is defined. We say that  F is defined at  A if a  F is a function restricted to the member  A of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F defAt  A  <->  ( A  e.  dom 
 F  /\  Fun  ( F  |`  { A } )
 ) )
 
Definitiondf-afv 27386* Alternative definition of the value of a function,  ( F''' A ), also known as function application. In contrast to  ( F `  A )  =  (/) (see df-fv 5228 and ndmfv 5513),  ( F''' A )  =  _V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F''' A )  =  if ( F defAt  A ,  U. { x  |  ( F
 " { A }
 )  =  { x } } ,  _V )
 
Definitiondf-aov 27387 Define the value of an operation. In contrast to df-ov 5822, the alternative definition for a function value ( see df-afv 27386) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation  F and its arguments  A and  B- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |- (( A F B))  =  ( F'''
 <. A ,  B >. )
 
18.23.2.1  Restricted quantification (extension)
 
Theoremralbinrald 27388* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
 |-  ( ph  ->  X  e.  A )   &    |-  ( x  e.  A  ->  x  =  X )   &    |-  ( x  =  X  ->  ( ps  <->  th ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 th ) )
 
18.23.2.2  The universal class (extension)
 
Theoremnvelim 27389 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4153. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( A  =  _V  ->  -.  A  e.  B )
 
18.23.2.3  Relations (extension)
 
Theoremsbcrel 27390 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ].
 Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
 
Theoremcsbdmg 27391 Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )
 
Theoremdmmpt2g 27392* Domain of a class given by the "maps to" notation, closed form of dmmpt2 6155. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( C  e.  V  ->  dom  F  =  ( A  X.  B ) )
 
Theoremeldmressn 27393 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )
 
Theoremdmressnsn 27394 The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( A  e.  dom  F  ->  dom  ( F  |`  { A } )  =  { A } )
 
Theoremeldmressnsn 27395 The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( A  e.  dom  F  ->  A  e.  dom  ( F  |` 
 { A } )
 )
 
18.23.2.4  Functions (extension)
 
Theoremsbcfun 27396 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ].
 Fun  F  <->  Fun  [_ A  /  x ]_ F ) )
 
Theoremfvfundmfvn0 27397 If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
 
Theoremfveqvfvv 27398 If a function's value at an argument is the universal class (which can never be the case because of fvex 5499), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything ( see pm2.21i 123). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F `  A )  =  _V  ->  ( F `  A )  =  B )
 
Theoremfunresfunco 27399 Composition of two functions, generalization of funco 5257. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( Fun  ( F  |` 
 ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
 
Theoremfnresfnco 27400 Composition of two functions, similar to fnco 5317. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( F  |`  ran  G )  Fn  ran  G  /\  G  Fn  B )  ->  ( F  o.  G )  Fn  B )
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