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Theorem bdreu 9975
 Description: Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula need not be bounded even if and are. Indeed, is bounded by bdcvv 9977, and (in minimal propositional calculus), so by bd0 9944, if were bounded when is bounded, then would be bounded as well when is bounded, which is not the case. The same remark holds with . (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdreu.1 BOUNDED
Assertion
Ref Expression
bdreu BOUNDED
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem bdreu
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bdreu.1 . . . 4 BOUNDED
21ax-bdex 9939 . . 3 BOUNDED
3 ax-bdeq 9940 . . . . . 6 BOUNDED
41, 3ax-bdim 9934 . . . . 5 BOUNDED
54ax-bdal 9938 . . . 4 BOUNDED
65ax-bdex 9939 . . 3 BOUNDED
72, 6ax-bdan 9935 . 2 BOUNDED
8 reu3 2731 . 2
97, 8bd0r 9945 1 BOUNDED
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97  wral 2306  wrex 2307  wreu 2308  BOUNDED wbd 9932 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdim 9934  ax-bdan 9935  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033  df-clel 2036  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314 This theorem is referenced by:  bdrmo  9976
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