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Theorem bdreu 11746
Description: Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula  A. x  e.  A ph need not be bounded even if 
A and  ph are. Indeed,  _V is bounded by bdcvv 11748, and  |-  ( A. x  e. 
_V ph  <->  A. x ph ) (in minimal propositional calculus), so by bd0 11715, if  A. x  e. 
_V ph were bounded when  ph is bounded, then  A. x ph would be bounded as well when  ph is bounded, which is not the case. The same remark holds with  E. ,  E! ,  E*. (Contributed by BJ, 16-Oct-2019.)

Hypothesis
Ref Expression
bdreu.1  |- BOUNDED  ph
Assertion
Ref Expression
bdreu  |- BOUNDED  E! x  e.  y 
ph
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bdreu
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bdreu.1 . . . 4  |- BOUNDED  ph
21ax-bdex 11710 . . 3  |- BOUNDED  E. x  e.  y 
ph
3 ax-bdeq 11711 . . . . . 6  |- BOUNDED  x  =  z
41, 3ax-bdim 11705 . . . . 5  |- BOUNDED  ( ph  ->  x  =  z )
54ax-bdal 11709 . . . 4  |- BOUNDED  A. x  e.  y  ( ph  ->  x  =  z )
65ax-bdex 11710 . . 3  |- BOUNDED  E. z  e.  y 
A. x  e.  y  ( ph  ->  x  =  z )
72, 6ax-bdan 11706 . 2  |- BOUNDED  ( E. x  e.  y  ph  /\  E. z  e.  y  A. x  e.  y  ( ph  ->  x  =  z ) )
8 reu3 2805 . 2  |-  ( E! x  e.  y  ph  <->  ( E. x  e.  y 
ph  /\  E. z  e.  y  A. x  e.  y  ( ph  ->  x  =  z ) ) )
97, 8bd0r 11716 1  |- BOUNDED  E! x  e.  y 
ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wral 2359   E.wrex 2360   E!wreu 2361  BOUNDED wbd 11703
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11704  ax-bdim 11705  ax-bdan 11706  ax-bdal 11709  ax-bdex 11710  ax-bdeq 11711
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367
This theorem is referenced by:  bdrmo  11747
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