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Theorem bj-axun2 10849
Description: axun2 4192 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 10755 . . . 4  |- BOUNDED  z  e.  w
21ax-bdex 10753 . . 3  |- BOUNDED  E. w  e.  x  z  e.  w
3 df-rex 2355 . . . 4  |-  ( E. w  e.  x  z  e.  w  <->  E. w
( w  e.  x  /\  z  e.  w
) )
4 exancom 1540 . . . 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 10758 . 2  |- BOUNDED  E. w ( z  e.  w  /\  w  e.  x )
7 ax-un 4190 . 2  |-  E. y A. z ( E. w
( z  e.  w  /\  w  e.  x
)  ->  z  e.  y )
86, 7bdbm1.3ii 10825 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 1283   E.wex 1422   E.wrex 2350
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-un 4190  ax-bd0 10747  ax-bdex 10753  ax-bdel 10755  ax-bdsep 10818
This theorem depends on definitions:  df-bi 115  df-rex 2355
This theorem is referenced by: (None)
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