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Metamath Proof Explorer

Theorem sbtr

Description: 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.)

	Ref	Expression
Hypotheses	sbtr.nf	$⊢ Ⅎ y φ$
	sbtr.1	$⊢ [y/ x] φ$
Assertion	sbtr	$⊢ φ$

Proof

Step	Hyp	Ref	Expression
1		sbtr.nf	$⊢ Ⅎ y φ$
2		sbtr.1	$⊢ [y/ x] φ$
3	1	sbtrt	$⊢ \forall y [y/ x] φ \to φ$
4	3 2	mpg	$⊢ φ$