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Theorem sbtr 2554
 Description: A partial converse to sbt 2552. 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 𝑦𝜑
sbtr.1 [𝑦 / 𝑥]𝜑
Assertion
Ref Expression
sbtr 𝜑

Proof of Theorem sbtr
StepHypRef Expression
1 sbtr.nf . . 3 𝑦𝜑
21sbtrt 2553 . 2 (∀𝑦[𝑦 / 𝑥]𝜑𝜑)
3 sbtr.1 . 2 [𝑦 / 𝑥]𝜑
42, 3mpg 1898 1 𝜑
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnf 1884  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-12 2222  ax-13 2391 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070 This theorem is referenced by: (None)
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