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Theorem bj-axun2 11236
Description: axun2 4235 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 11142 . . . 4  |- BOUNDED  z  e.  w
21ax-bdex 11140 . . 3  |- BOUNDED  E. w  e.  x  z  e.  w
3 df-rex 2361 . . . 4  |-  ( E. w  e.  x  z  e.  w  <->  E. w
( w  e.  x  /\  z  e.  w
) )
4 exancom 1542 . . . 4  |-  ( E. w ( w  e.  x  /\  z  e.  w )  <->  E. w
( z  e.  w  /\  w  e.  x
) )
53, 4bitri 182 . . 3  |-  ( E. w  e.  x  z  e.  w  <->  E. w
( z  e.  w  /\  w  e.  x
) )
62, 5bd0 11145 . 2  |- BOUNDED  E. w ( z  e.  w  /\  w  e.  x )
7 ax-un 4233 . 2  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
86, 7bdbm1.3ii 11212 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 102    <-> wb 103   A.wal 1285   E.wex 1424   E.wrex 2356
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-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-un 4233  ax-bd0 11134  ax-bdex 11140  ax-bdel 11142  ax-bdsep 11205
This theorem depends on definitions:  df-bi 115  df-rex 2361
This theorem is referenced by: (None)
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