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Theorem bdzfauscl 10966
Description: Closed form of the version of zfauscl 3918 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
Hypothesis
Ref Expression
bdzfauscl.bd  |- BOUNDED  ph
Assertion
Ref Expression
bdzfauscl  |-  ( A  e.  V  ->  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) )
Distinct variable groups:    x, y, A    ph, y
Allowed substitution hints:    ph( x)    V( x, y)

Proof of Theorem bdzfauscl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq2 2146 . . . . . 6  |-  ( z  =  A  ->  (
x  e.  z  <->  x  e.  A ) )
21anbi1d 453 . . . . 5  |-  ( z  =  A  ->  (
( x  e.  z  /\  ph )  <->  ( x  e.  A  /\  ph )
) )
32bibi2d 230 . . . 4  |-  ( z  =  A  ->  (
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )  <-> 
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) ) )
43albidv 1747 . . 3  |-  ( z  =  A  ->  ( A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)  <->  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) ) )
54exbidv 1748 . 2  |-  ( z  =  A  ->  ( E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)  <->  E. y A. x
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) ) )
6 bdzfauscl.bd . . 3  |- BOUNDED  ph
76bdsep1 10961 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
85, 7vtoclg 2667 1  |-  ( A  e.  V  ->  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434  BOUNDED wbd 10888
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bdsep 10960
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612
This theorem is referenced by:  bdinex1  10975
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