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Theorem bdnel 11402
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 𝑥𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem bdnel
StepHypRef Expression
1 bdnel.1 . . . 4 BOUNDED 𝐴
21bdeli 11394 . . 3 BOUNDED 𝑥𝐴
32ax-bdn 11365 . 2 BOUNDED ¬ 𝑥𝐴
4 df-nel 2351 . 2 (𝑥𝐴 ↔ ¬ 𝑥𝐴)
53, 4bd0r 11373 1 BOUNDED 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1438  wnel 2350  BOUNDED wbd 11360  BOUNDED wbdc 11388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-4 1445  ax-bd0 11361  ax-bdn 11365
This theorem depends on definitions:  df-bi 115  df-nel 2351  df-bdc 11389
This theorem is referenced by: (None)
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