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Theorem bdzfauscl 14524
Description: Closed form of the version of zfauscl 4123 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 2241 . . . . . 6  |-  ( z  =  A  ->  (
x  e.  z  <->  x  e.  A ) )
21anbi1d 465 . . . . 5  |-  ( z  =  A  ->  (
( x  e.  z  /\  ph )  <->  ( x  e.  A  /\  ph )
) )
32bibi2d 232 . . . 4  |-  ( z  =  A  ->  (
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )  <-> 
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) ) )
43albidv 1824 . . 3  |-  ( z  =  A  ->  ( A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)  <->  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
) ) )
54exbidv 1825 . 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 14519 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
85, 7vtoclg 2797 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 104    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148  BOUNDED wbd 14446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bdsep 14518
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  bdinex1  14533
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