Theorem sbtr 2554

Description: A partial converse to sbt 2067. 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 2386. (Contributed by BJ, 15-Sep-2018.) (New usage is discouraged.)

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 2553	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑)
3		sbtr.1	. 2 ⊢ [𝑦 / 𝑥]𝜑
4	2, 3	mpg 1794	1 ⊢ 𝜑

Syntax hints:  Ⅎwnf 1780  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066

This theorem is referenced by: (None)