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Theorem bj-axun2 12915
Description: axun2 4325 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axun2  |-  E. y A. z ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
Distinct variable group:    x, w, y, z

Proof of Theorem bj-axun2
StepHypRef Expression
1 ax-bdel 12821 . . . 4  |- BOUNDED  z  e.  w
21ax-bdex 12819 . . 3  |- BOUNDED  E. w  e.  x  z  e.  w
3 df-rex 2397 . . . 4  |-  ( E. w  e.  x  z  e.  w  <->  E. w
( w  e.  x  /\  z  e.  w
) )
4 exancom 1570 . . . 4  |-  ( E. w ( w  e.  x  /\  z  e.  w )  <->  E. w
( z  e.  w  /\  w  e.  x
) )
53, 4bitri 183 . . 3  |-  ( E. w  e.  x  z  e.  w  <->  E. w
( z  e.  w  /\  w  e.  x
) )
62, 5bd0 12824 . 2  |- BOUNDED  E. w ( z  e.  w  /\  w  e.  x )
7 ax-un 4323 . 2  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
86, 7bdbm1.3ii 12891 1  |-  E. y A. z ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1312   E.wex 1451   E.wrex 2392
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-un 4323  ax-bd0 12813  ax-bdex 12819  ax-bdel 12821  ax-bdsep 12884
This theorem depends on definitions:  df-bi 116  df-rex 2397
This theorem is referenced by: (None)
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