reserve T, X, Y for Subset of MC-wff;
reserve p, q, r, s for Element of MC-wff;

theorem Th37:
  ( p => ( q => r )) => (( p '&' q ) => r ) in IPC-Taut
proof
A1: (( p '&' q ) => q) => (( q => r ) => (( p '&' q ) => r )) in IPC-Taut by
Th24;
  ( p '&' q ) => q in IPC-Taut by Def14;
  then ( q => r ) => (( p '&' q ) => r ) in IPC-Taut by A1,Def14;
  then
A2: p => (( q => r ) => (( p '&' q ) => r )) in IPC-Taut by Th18;
A3: ( p => (( p '&' q ) => r )) => ((p '&' q ) => ( p => r )) in IPC-Taut by
Th29;
  p => (( q => r ) => (( p '&' q ) => r )) => ((p => ( q => r )) => ( p =>
  (( p '&' q ) => r ))) in IPC-Taut by Def14;
  then (p => ( q => r )) => ( p => (( p '&' q ) => r )) in IPC-Taut by A2,Def14
;
  then
A4: (p => ( q => r )) => ((p '&' q ) => ( p => r )) in IPC-Taut by A3,Th26;
A5: ( p '&' q ) => p in IPC-Taut by Def14;
  ((p '&' q ) => ( p => r )) => ((( p '&' q ) => p ) => (( p '&' q ) => r
  )) in IPC-Taut by Def14;
  then ((p '&' q ) => ( p => r )) => (( p '&' q ) => r ) in IPC-Taut by A5,Th32
;
  hence thesis by A4,Th26;
end;
