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Theorem sbtr 2558
Description: A partial converse to sbt 2071. 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 2391. (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 2557 . 2 (∀𝑦[𝑦 / 𝑥]𝜑𝜑)
3 sbtr.1 . 2 [𝑦 / 𝑥]𝜑
42, 3mpg 1799 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wnf 1785  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by: (None)
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