On Feb 4, 9:21 am, Charlie-Boo <shymath...@gmail.com> wrote: > On Feb 3, 5:29 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > On Feb 2, 3:38 am, Charlie-Boo <shymath...@gmail.com> wrote: > > > > That is, if for every wff w in system A there is a wff v in system B > > > such that |-w(x) iff |-v(x) for all x, and likewise for vice-versa B > > > is in A, then do systems A and B prove the same theorems? > > > > C-B > > > by extension system A = system B > > > However, one system may have restricted comprehension on WFF while the > > other does not. > > > Herc > > Are you saying yes to my original question?
Your subject line I'm ignoring.
(ALL(thm) thm e theoryA <-> thm e theoryB) <-> (theoryA = theoryB)
ALL(thm) A|-thm <-> B|-thm <-> A=B
However the set of WFF in each may be different.
The parameter X limits what you are trying to say as theorems have higher arity than 1.