Theorem sbtr 2409

Description: A partial converse to sbt 2407. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. (Contributed by BJ, 15-Sep-2018.)

Hypotheses

Ref	Expression
sbtr.nf	⊢ Ⅎ𝑦𝜑
sbtr.1	⊢ [𝑦 / 𝑥]𝜑

Assertion

Ref	Expression
sbtr	⊢ 𝜑

Proof of Theorem sbtr

Step	Hyp	Ref	Expression
1		sbtr.nf	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sbtrt 2408	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑)
3		sbtr.1	. 2 ⊢ [𝑦 / 𝑥]𝜑
4	2, 3	mpg 1715	1 ⊢ 𝜑

Syntax hints:  Ⅎwnf 1699  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by: (None)