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Theorem bdab 13120
Description: Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
Hypothesis
Ref Expression
bdab.1 BOUNDED 𝜑
Assertion
Ref Expression
bdab BOUNDED 𝑥 ∈ {𝑦𝜑}

Proof of Theorem bdab
StepHypRef Expression
1 bdab.1 . . 3 BOUNDED 𝜑
21ax-bdsb 13104 . 2 BOUNDED [𝑥 / 𝑦]𝜑
3 df-clab 2126 . 2 (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
42, 3bd0r 13107 1 BOUNDED 𝑥 ∈ {𝑦𝜑}
Colors of variables: wff set class
Syntax hints:  wcel 1480  [wsb 1735  {cab 2125  BOUNDED wbd 13094
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-bd0 13095  ax-bdsb 13104
This theorem depends on definitions:  df-bi 116  df-clab 2126
This theorem is referenced by:  bdcab  13131  bdsbcALT  13141
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