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	|- F/ y ph
	sbtr.1	|- [ y / x ] ph
Assertion	sbtr	|- ph

Proof

Step	Hyp	Ref	Expression
1		sbtr.nf	|- F/ y ph
2		sbtr.1	|- [ y / x ] ph
3	1	sbtrt	|- ( A. y [ y / x ] ph -> ph )
4	3 2	mpg	|- ph