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Theorem bdreu 14229
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 14231, and  |-  ( A. x  e. 
_V ph  <->  A. x ph ) (in minimal propositional calculus), so by bd0 14198, 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 14193 . . 3  |- BOUNDED  E. x  e.  y 
ph
3 ax-bdeq 14194 . . . . . 6  |- BOUNDED  x  =  z
41, 3ax-bdim 14188 . . . . 5  |- BOUNDED  ( ph  ->  x  =  z )
54ax-bdal 14192 . . . 4  |- BOUNDED  A. x  e.  y  ( ph  ->  x  =  z )
65ax-bdex 14193 . . 3  |- BOUNDED  E. z  e.  y 
A. x  e.  y  ( ph  ->  x  =  z )
72, 6ax-bdan 14189 . 2  |- BOUNDED  ( E. x  e.  y  ph  /\  E. z  e.  y  A. x  e.  y  ( ph  ->  x  =  z ) )
8 reu3 2927 . 2  |-  ( E! x  e.  y  ph  <->  ( E. x  e.  y 
ph  /\  E. z  e.  y  A. x  e.  y  ( ph  ->  x  =  z ) ) )
97, 8bd0r 14199 1  |- BOUNDED  E! x  e.  y 
ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wral 2455   E.wrex 2456   E!wreu 2457  BOUNDED wbd 14186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14187  ax-bdim 14188  ax-bdan 14189  ax-bdal 14192  ax-bdex 14193  ax-bdeq 14194
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463
This theorem is referenced by:  bdrmo  14230
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