|Description: Alternate definition of
proper substitution. Note that the occurrences
of a given variable are either all bound (𝑥, 𝑦) or all free
(𝑡). Also note that the definiens uses
only primitive symbols.
It is obtained by applying twice Tarski's definition sb6 2457
valid for disjoint variables, so we introduce a dummy variable 𝑦 to
isolate 𝑥 from 𝑡, as in dfsb7 2483 with respect to sb5 2458.
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 32743, 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 2091, of the substitution axiom ax-12 2087,
and of commutation of quantifiers ax-11 2074; that is, ax-13 2282 will often
be avoided. (Contributed by BJ,