Theorem sbtr

Description: A partial converse to sbt . If the substitution of a variable for a non-free 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.)

	Ref	Expression
Hypotheses	sbtr.nf	$$⊢Ⅎy\phi$$
	sbtr.1	$$⊢\left[y/x\right]\phi$$
Assertion	sbtr	$$⊢\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbtr.nf	$$⊢Ⅎy\phi$$
2		sbtr.1	$$⊢\left[y/x\right]\phi$$
3	1	sbtrt	$$⊢\forall y\left[y/x\right]\phi \to \phi$$
4	3 2	mpg	$$⊢\phi$$