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Theorem bdrmo 14264
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 14227 . . 3 BOUNDED𝑥𝑦 𝜑
31bdreu 14263 . . 3 BOUNDED ∃!𝑥𝑦 𝜑
42, 3ax-bdim 14222 . 2 BOUNDED (∃𝑥𝑦 𝜑 → ∃!𝑥𝑦 𝜑)
5 rmo5 2692 . 2 (∃*𝑥𝑦 𝜑 ↔ (∃𝑥𝑦 𝜑 → ∃!𝑥𝑦 𝜑))
64, 5bd0r 14233 1 BOUNDED ∃*𝑥𝑦 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wrex 2456  ∃!wreu 2457  ∃*wrmo 2458  BOUNDED wbd 14220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14221  ax-bdim 14222  ax-bdan 14223  ax-bdal 14226  ax-bdex 14227  ax-bdeq 14228
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463
This theorem is referenced by: (None)
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