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Theorem bdreu 10804
Description: Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula 𝑥𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 10806, and (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 10773, if 𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then 𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)

Hypothesis
Ref Expression
bdreu.1 BOUNDED 𝜑
Assertion
Ref Expression
bdreu BOUNDED ∃!𝑥𝑦 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bdreu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdreu.1 . . . 4 BOUNDED 𝜑
21ax-bdex 10768 . . 3 BOUNDED𝑥𝑦 𝜑
3 ax-bdeq 10769 . . . . . 6 BOUNDED 𝑥 = 𝑧
41, 3ax-bdim 10763 . . . . 5 BOUNDED (𝜑𝑥 = 𝑧)
54ax-bdal 10767 . . . 4 BOUNDED𝑥𝑦 (𝜑𝑥 = 𝑧)
65ax-bdex 10768 . . 3 BOUNDED𝑧𝑦𝑥𝑦 (𝜑𝑥 = 𝑧)
72, 6ax-bdan 10764 . 2 BOUNDED (∃𝑥𝑦 𝜑 ∧ ∃𝑧𝑦𝑥𝑦 (𝜑𝑥 = 𝑧))
8 reu3 2783 . 2 (∃!𝑥𝑦 𝜑 ↔ (∃𝑥𝑦 𝜑 ∧ ∃𝑧𝑦𝑥𝑦 (𝜑𝑥 = 𝑧)))
97, 8bd0r 10774 1 BOUNDED ∃!𝑥𝑦 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wral 2349  wrex 2350  ∃!wreu 2351  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:  bdrmo  10805
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