Theorem sbtr

A partial converse to sbt. If the substitution of a variable for a nonfree one in a wff gives a theorem, then the original wff is a theorem. Usage of this theorem is discouraged because it depends on [ax-13]. (Contributed by BJ, 15-Sep-2018.) (New usage is discouraged.)

Hypotheses
sbtr.nf
\( \vdash \Finv y \, \varphi \)
sbtr.1
\( \vdash [ y / x ] \, \varphi \)
Conclusion
\( \vdash \varphi \)
Proof
sbtr.nf
l1
sbtrt _
\( \vdash \forall y \, [ y / x ] \, \varphi \rightarrow \varphi \)
sbtr.1
\( \vdash [ y / x ] \, \varphi \)
mpg l1 _