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Theorem bdph 9285
Description: A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
Hypothesis
Ref Expression
bdph.1 BOUNDED {xφ}
Assertion
Ref Expression
bdph BOUNDED φ

Proof of Theorem bdph
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 bdph.1 . . . . 5 BOUNDED {xφ}
21bdeli 9281 . . . 4 BOUNDED y {xφ}
3 df-clab 2024 . . . 4 (y {xφ} ↔ [y / x]φ)
42, 3bd0 9259 . . 3 BOUNDED [y / x]φ
54ax-bdsb 9257 . 2 BOUNDED [x / y][y / x]φ
6 sbid2v 1869 . 2 ([x / y][y / x]φφ)
75, 6bd0 9259 1 BOUNDED φ
Colors of variables: wff set class
Syntax hints:   wcel 1390  [wsb 1642  {cab 2023  BOUNDED wbd 9247  BOUNDED wbdc 9275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-bd0 9248  ax-bdsb 9257
This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-bdc 9276
This theorem is referenced by:  bds  9286
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