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Theorem sbtr 2511
Description: A partial converse to sbt 2062. 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 2367. (Contributed by BJ, 15-Sep-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbtr.nf 𝑦𝜑
sbtr.1 [𝑦 / 𝑥]𝜑
Assertion
Ref Expression
sbtr 𝜑

Proof of Theorem sbtr
StepHypRef Expression
1 sbtr.nf . . 3 𝑦𝜑
21sbtrt 2510 . 2 (∀𝑦[𝑦 / 𝑥]𝜑𝜑)
3 sbtr.1 . 2 [𝑦 / 𝑥]𝜑
42, 3mpg 1792 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wnf 1778  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167  ax-13 2367
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061
This theorem is referenced by: (None)
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