Article SCHEMS_1, MML version 4.99.1005

:: SCHEMS_1:sch 1
scheme SCHEMS_1:sch 1
ex b1 being set st
   P1[b1]
provided
   for b1 being set holds
      P1[b1];


:: SCHEMS_1:sch 2
scheme SCHEMS_1:sch 2
for b1 being set holds
   ex b2 being set st
      P1[b2, b1]
provided
   ex b1 being set st
      for b2 being set holds
         P1[b1, b2];


:: SCHEMS_1:sch 3
scheme SCHEMS_1:sch 3
(for b1 being set holds
   P1[b1] implies for b1 being set holds
   P2[b1])
provided
   for b1 being set
         st P1[b1]
      holds P2[b1];


:: SCHEMS_1:sch 4
scheme SCHEMS_1:sch 4
   (for b1 being set holds
      P1[b1]
iff
   for b1 being set holds
      P2[b1])
provided
   for b1 being set holds
         P1[b1]
      iff
         P2[b1];


:: SCHEMS_1:sch 5
scheme SCHEMS_1:sch 5
(for b1 being set holds
   P1[b1] implies P2[])
provided
   for b1 being set
         st P1[b1]
      holds P2[];


:: SCHEMS_1:sch 6
scheme SCHEMS_1:sch 6
ex b1 being set st
   for b2 being set
         st not (P1[b1])
      holds P2[b2]
provided
   (for b1 being set holds
      not (P1[b1]) implies for b1 being set holds
      P2[b1]);


:: SCHEMS_1:sch 7
scheme SCHEMS_1:sch 7
(for b1 being set holds
   not (P1[b1]) implies for b1 being set holds
   P2[b1])
provided
   ex b1 being set st
      for b2 being set
            st not (P1[b1])
         holds P2[b2];


:: SCHEMS_1:sch 8
scheme SCHEMS_1:sch 8
ex b1 being set st
   for b2 being set
         st not (P1[b1])
      holds P2[b2]
provided
   for b1 being set holds
      ex b2 being set st
         (P1[b2] or P2[b1]);


:: SCHEMS_1:sch 9
scheme SCHEMS_1:sch 9
for b1 being set holds
   ex b2 being set st
      P1[b2] & P2[b1]
provided
   (ex b1 being set st
       P1[b1]) &
    (for b1 being set holds
       P2[b1]);


:: SCHEMS_1:sch 10
scheme SCHEMS_1:sch 10
(ex b1 being set st
    P1[b1]) &
 (for b1 being set holds
    P2[b1])
provided
   for b1 being set holds
      ex b2 being set st
         P1[b2] & P2[b1];


:: SCHEMS_1:sch 11
scheme SCHEMS_1:sch 11
ex b1 being set st
   for b2 being set holds
      P1[b1] & P2[b2]
provided
   for b1 being set holds
      ex b2 being set st
         P1[b2] & P2[b1];


:: SCHEMS_1:sch 12
scheme SCHEMS_1:sch 12
ex b1 being set st
   for b2 being set holds
      P1[b2, b1]
provided
   for b1, b2 being set holds
   P1[b1, b2];


:: SCHEMS_1:sch 13
scheme SCHEMS_1:sch 13
ex b1 being set st
   P1[b1, b1]
provided
   ex b1 being set st
      for b2 being set holds
         P1[b1, b2];


:: SCHEMS_1:sch 14
scheme SCHEMS_1:sch 14
for b1 being set holds
   ex b2 being set st
      P1[b2, b1]
provided
   for b1 being set holds
      P1[b1, b1];


:: SCHEMS_1:sch 15
scheme SCHEMS_1:sch 15
for b1 being set holds
   ex b2 being set st
      P1[b1, b2]
provided
   for b1 being set holds
      P1[b1, b1];


:: SCHEMS_1:sch 16
scheme SCHEMS_1:sch 16
ex b1, b2 being set st
   P1[b1, b2]
provided
   for b1 being set holds
      ex b2 being set st
         P1[b2, b1];