Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdreu Structured version   GIF version

Theorem bdreu 9310
 Description: Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula ∀x ∈ Aφ need not be bounded even if A and φ are. Indeed, V is bounded by bdcvv 9312, and ⊢ (∀x ∈ Vφ ↔ ∀xφ) (in minimal propositional calculus), so by bd0 9279, 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 ∃!x y 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 9274 . . 3 BOUNDED x y φ
3 ax-bdeq 9275 . . . . . 6 BOUNDED x = z
41, 3ax-bdim 9269 . . . . 5 BOUNDED (φx = z)
54ax-bdal 9273 . . . 4 BOUNDED x y (φx = z)
65ax-bdex 9274 . . 3 BOUNDED z y x y (φx = z)
72, 6ax-bdan 9270 . 2 BOUNDED (x y φ z y x y (φx = z))
8 reu3 2725 . 2 (∃!x y φ ↔ (x y φ z y x y (φx = z)))
97, 8bd0r 9280 1 BOUNDED ∃!x y φ
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wral 2300  ∃wrex 2301  ∃!wreu 2302  BOUNDED wbd 9267 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9268  ax-bdim 9269  ax-bdan 9270  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308 This theorem is referenced by:  bdrmo  9311
 Copyright terms: Public domain W3C validator