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Theorem bdreu 12855
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 12857, and (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 12824, 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 12819 . . 3 BOUNDED𝑥𝑦 𝜑
3 ax-bdeq 12820 . . . . . 6 BOUNDED 𝑥 = 𝑧
41, 3ax-bdim 12814 . . . . 5 BOUNDED (𝜑𝑥 = 𝑧)
54ax-bdal 12818 . . . 4 BOUNDED𝑥𝑦 (𝜑𝑥 = 𝑧)
65ax-bdex 12819 . . 3 BOUNDED𝑧𝑦𝑥𝑦 (𝜑𝑥 = 𝑧)
72, 6ax-bdan 12815 . 2 BOUNDED (∃𝑥𝑦 𝜑 ∧ ∃𝑧𝑦𝑥𝑦 (𝜑𝑥 = 𝑧))
8 reu3 2845 . 2 (∃!𝑥𝑦 𝜑 ↔ (∃𝑥𝑦 𝜑 ∧ ∃𝑧𝑦𝑥𝑦 (𝜑𝑥 = 𝑧)))
97, 8bd0r 12825 1 BOUNDED ∃!𝑥𝑦 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2391  wrex 2392  ∃!wreu 2393  BOUNDED wbd 12812
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12813  ax-bdim 12814  ax-bdan 12815  ax-bdal 12818  ax-bdex 12819  ax-bdeq 12820
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-cleq 2108  df-clel 2111  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399
This theorem is referenced by:  bdrmo  12856
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