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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	⊢ ( ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 )
4	3 2	mpg	⊢ 𝜑