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Theorem bdnel 13082
Description: Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdnel.1  |- BOUNDED  A
Assertion
Ref Expression
bdnel  |- BOUNDED  x  e/  A
Distinct variable group:    x, A

Proof of Theorem bdnel
StepHypRef Expression
1 bdnel.1 . . . 4  |- BOUNDED  A
21bdeli 13074 . . 3  |- BOUNDED  x  e.  A
32ax-bdn 13045 . 2  |- BOUNDED  -.  x  e.  A
4 df-nel 2404 . 2  |-  ( x  e/  A  <->  -.  x  e.  A )
53, 4bd0r 13053 1  |- BOUNDED  x  e/  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1480    e/ wnel 2403  BOUNDED wbd 13040  BOUNDED wbdc 13068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-4 1487  ax-bd0 13041  ax-bdn 13045
This theorem depends on definitions:  df-bi 116  df-nel 2404  df-bdc 13069
This theorem is referenced by: (None)
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