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Theorem bdrmo 10805
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 10768 . . 3 BOUNDED𝑥𝑦 𝜑
31bdreu 10804 . . 3 BOUNDED ∃!𝑥𝑦 𝜑
42, 3ax-bdim 10763 . 2 BOUNDED (∃𝑥𝑦 𝜑 → ∃!𝑥𝑦 𝜑)
5 rmo5 2570 . 2 (∃*𝑥𝑦 𝜑 ↔ (∃𝑥𝑦 𝜑 → ∃!𝑥𝑦 𝜑))
64, 5bd0r 10774 1 BOUNDED ∃*𝑥𝑦 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wrex 2350  ∃!wreu 2351  ∃*wrmo 2352  BOUNDED wbd 10761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bd0 10762  ax-bdim 10763  ax-bdan 10764  ax-bdal 10767  ax-bdex 10768  ax-bdeq 10769
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-cleq 2075  df-clel 2078  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357
This theorem is referenced by: (None)
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