Description: Alternate definition of
proper substitution. Note that the occurrences
of a given variable are either all bound (   ) or all free
( ). Also note
that the definiens uses only primitive symbols.
It is obtained by applying twice Tarski's definition sb6 2429
which is
valid for disjoint variables, so we introduce a dummy variable to
isolate from
, as in dfsb7 2455 with respect to sb5 2430.
This double level definition will make several proofs using it appear as
doubled. Alternately, one could often first prove as a lemma the same
theorem with a DV condition on the substitute and the substituted
variables, and then prove the original theorem by applying this lemma
twice in a row.
This definition uses a dummy variable, but the justification theorem,
bj-ssbjust 32618, is provable from Tarski's FOL.
Once this is proved, more of the fundamental properties of proper
substitution will be provable from Tarski's FOL system, sometimes with
the help of specialization sp 2053, of the substitution axiom ax-12 2047,
and of commutation of quantifiers ax-11 2034; that is, ax-13 2246 will often
be avoided. (Contributed by BJ,
22-Dec-2020.) |
[
