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Theorem bdzfauscl 9883
Description: Closed form of the version of zfauscl 3874 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 2101 . . . . . 6  |-  ( z  =  A  ->  (
x  e.  z  <->  x  e.  A ) )
21anbi1d 438 . . . . 5  |-  ( z  =  A  ->  (
( x  e.  z  /\  ph )  <->  ( x  e.  A  /\  ph )
) )
32bibi2d 221 . . . 4  |-  ( z  =  A  ->  (
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )  <-> 
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) ) )
43albidv 1705 . . 3  |-  ( z  =  A  ->  ( A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)  <->  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) ) )
54exbidv 1706 . 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 9878 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
85, 7vtoclg 2610 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 97    <-> wb 98   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393  BOUNDED wbd 9805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 9877
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556
This theorem is referenced by:  bdinex1  9892
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