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Theorem bdrmo 13054
 Description: Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdrmo.1 BOUNDED 𝜑
Assertion
Ref Expression
bdrmo BOUNDED ∃*𝑥𝑦 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bdrmo
StepHypRef Expression
1 bdrmo.1 . . . 4 BOUNDED 𝜑
21ax-bdex 13017 . . 3 BOUNDED𝑥𝑦 𝜑
31bdreu 13053 . . 3 BOUNDED ∃!𝑥𝑦 𝜑
42, 3ax-bdim 13012 . 2 BOUNDED (∃𝑥𝑦 𝜑 → ∃!𝑥𝑦 𝜑)
5 rmo5 2646 . 2 (∃*𝑥𝑦 𝜑 ↔ (∃𝑥𝑦 𝜑 → ∃!𝑥𝑦 𝜑))
64, 5bd0r 13023 1 BOUNDED ∃*𝑥𝑦 𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wrex 2417  ∃!wreu 2418  ∃*wrmo 2419  BOUNDED wbd 13010 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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13011  ax-bdim 13012  ax-bdan 13013  ax-bdal 13016  ax-bdex 13017  ax-bdeq 13018 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424 This theorem is referenced by: (None)
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