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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmdandyv7 27901 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 27902 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 27903 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 27904 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 27905 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 27906 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 27907 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 27908 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 27909 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 27910 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 27911 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 27912 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 27913 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 27914 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 27915 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 27916 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 27917 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 27918 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 27919 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 27920 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 27921 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 27922 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 27923 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 27924 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 27925 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 27926 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 27927 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 27928 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 27929 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 27930 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 27931 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 27932 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 27933 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 27934 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 27935 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 27936 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 27937 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 27938 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 27939 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 27940 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 27941 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 27942 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 27943

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 27944 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 27945 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 27946* 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 27947* 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 27948* An equivalent expression for double restricted existence, analogous to rexsb 27946. (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 27949* 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 27950* 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 27951* 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 27952 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 27953* 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 27954 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 27955 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 27956 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 27957* 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 27958* 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 27959* 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 27960* 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 27961* 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 27962* 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 27963* 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 27964* 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 27965* 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 27966* 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 27967* 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 27968). (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 27968* 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 27969* 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 27970* 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 27969. (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 5263) assures that this values is always a set, see fex 5749. 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 5552 and fvprc 5519).

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 5553).

To avoid such an ambiguity, an alternative definition  ( F''' A ) ( see df-afv 27975) would be possible which evaluates to the universal class ( ( F''' A )  =  _V) if it is not meaningful (see afvnfundmuv 28002, ndmafv 28003, afvprc 28007 and nfunsnafv 28005), and which corresponds to the current definition ( ( F `  A )  =  ( F''' A )) if it is (see afvfundmfveq 28001). That means  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) (see afvpcfv0 28009).

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 dffv4 5522 of 
( F `  A
), we see that analogons for the following 6 theorems can be proven using the alternative definition: fveq1 5524-> afveq1 27997, fveq2 5525-> afveq2 27998, nffv 5532-> nfafv 27999, csbfv12g 5535-> csbafv12g , fvres 5542-> afvres 28034, fvco2 5594-> afvco2 28037. From these, only afvco2 28037 is a little bit tricky to be proven. 3 theorems proved by directly using dffv4 5522 are deprecated (usage discouraged) or within mathboxes: csbfv12gALT 5536, repfuntw 25160, csbfv12gALTVD 28675. 2 additional theorems proved by directly using dffv4 5522 can also be neclected: dffv3 5521 (used only in dffv4 5522 which is contained in a mathbox) and avril1 20836 (not used ;-)). However, the remaining 4 theorems proved by directly using dffv4 5522 are used more or less often: fv2 5520 (see below), fvprc 5519 (used in 127 proofs), fvex 5539 (used in about 1750 proofs), shftval 11569 (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 5520, for example, there are 3 theorems being proved by using fv2 5520: elfv 5523, df-fv 5263 and ovtpos 6249. While ovtpos 6249 is used in 14 proofs, elfv 5523 is used (indirectly via tz6.12-2 5516, tz6.12i 5548, and fvbr0 5549) for fvrn0 5550 (used in 18 proofs) , dcomex 8073 (used in 4 proofs) and ndmfv 5552 (used in 86 proofs). fv3 5541, which is also proven using elfv 5523, is only used by tz6.12-1 5544 which could be replaced by tz6.12-afv 28035! Finally, regarding elfv 5523, the proof of nfunsn 5558 uses tz6.12-2 5516, but nfunsn 5558 is only used for dffv2 5592 which itself is not used in any proof. The remaining theorem df-fv 5263 whose proof uses fv2 5520 is (indirectly) used for adjbdln 22663 (used in 7 proofs ), isum 12192 (used in 7 proofs ), sum0 12194 (used in 42 proofs ), fsumser 12203 (used in 40 proofs ) and logtayl 20007 (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 27976.

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

 
Syntaxwdfat 27971 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 27972 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 5263), 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 27995, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5263 and df-ima 4702. 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 27973 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 5861.
 class (( A F B))
 
Definitiondf-dfat 27974 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 27975* Alternative definition of the value of a function,  ( F''' A ), also known as function application. In contrast to  ( F `  A )  =  (/) (see df-fv 5263 and ndmfv 5552),  ( 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 27976 Define the value of an operation. In contrast to df-ov 5861, the alternative definition for a function value ( see df-afv 27975) 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 27977* 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 27978 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 27979 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 27980 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 27981* Domain of a class given by the "maps to" notation, closed form of dmmpt2 6194. (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 27982 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 27983 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 27984 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 27985 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 27986 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 27987 If a function's value at an argument is the universal class (which can never be the case because of fvex 5539), 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 27988 Composition of two functions, generalization of funco 5292. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( Fun  ( F  |` 
 ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
 
Theoremfnresfnco 27989 Composition of two functions, similar to fnco 5352. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( F  |`  ran  G )  Fn  ran  G  /\  G  Fn  B )  ->  ( F  o.  G )  Fn  B )
 
Theoremfuncoressn 27990 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( ( G `
  X )  e. 
 dom  F  /\  Fun  ( F  |`  { ( G `
  X ) }
 ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( ( F  o.  G )  |`  { X } ) )
 
Theoremfunressnfv 27991 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( X  e.  dom  ( F  o.  G )  /\  Fun  ( ( F  o.  G )  |`  { X } ) ) 
 /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( F  |`  { ( G `
  X ) }
 ) )
 
18.23.2.5  Predicate "defined at"
 
Theoremdfateq12d 27992 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
 
Theoremnfdfat 27993 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g. for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |- 
 F/ x  F defAt  A
 
Theoremdfdfat2 27994* Alternate definition of the predicate "defined at" not using the  Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F defAt  A  <->  ( A  e.  dom 
 F  /\  E! y  A F y ) )
 
18.23.2.6  Alternative definition of the value of a function
 
Theoremdfafv2 27995 Alternative definition of  ( F''' A ) using  ( F `
 A ) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F''' A )  =  if ( F defAt  A ,  ( F `  A ) ,  _V )
 
Theoremafveq12d 27996 Equality deduction for function value, analogous to fveq12d 5531. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
 
Theoremafveq1 27997 Equality theorem for function value, analogous to fveq1 5524. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F  =  G  ->  ( F''' A )  =  ( G''' A ) )
 
Theoremafveq2 27998 Equality theorem for function value, analogous to fveq1 5524. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  =  B  ->  ( F''' A )  =  ( F''' B ) )
 
Theoremnfafv 27999 Bound-variable hypothesis builder for function value, analogous to nffv 5532. To prove a deduction version of this analogous to nffvd 5534 is not easily possible because a deduction version of nfdfat 27993 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F''' A )
 
Theoremcsbafv12g 28000 Move class substitution in and out of a function value, analogous to csbfv12g 5535, with a direct proof proposed by Mario Carneiro, analogous to csbovg 5889. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
 [_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
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