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Theorem zfrep4 4576
Description: A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
Hypotheses
Ref Expression
zfrep4.1  |-  { x  |  ph }  e.  _V
zfrep4.2  |-  ( ph  ->  E. z A. y
( ps  ->  y  =  z ) )
Assertion
Ref Expression
zfrep4  |-  { y  |  E. x (
ph  /\  ps ) }  e.  _V
Distinct variable groups:    ph, y, z    ps, z    x, y, z
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem zfrep4
StepHypRef Expression
1 abid 2444 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
21anbi1i 695 . . . 4  |-  ( ( x  e.  { x  |  ph }  /\  ps ) 
<->  ( ph  /\  ps ) )
32exbii 1668 . . 3  |-  ( E. x ( x  e. 
{ x  |  ph }  /\  ps )  <->  E. x
( ph  /\  ps )
)
43abbii 2591 . 2  |-  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }  =  {
y  |  E. x
( ph  /\  ps ) }
5 nfab1 2621 . . . . 5  |-  F/_ x { x  |  ph }
6 zfrep4.1 . . . . 5  |-  { x  |  ph }  e.  _V
7 zfrep4.2 . . . . . 6  |-  ( ph  ->  E. z A. y
( ps  ->  y  =  z ) )
81, 7sylbi 195 . . . . 5  |-  ( x  e.  { x  | 
ph }  ->  E. z A. y ( ps  ->  y  =  z ) )
95, 6, 8zfrepclf 4574 . . . 4  |-  E. z A. y ( y  e.  z  <->  E. x ( x  e.  { x  | 
ph }  /\  ps ) )
10 abeq2 2581 . . . . 5  |-  ( z  =  { y  |  E. x ( x  e.  { x  | 
ph }  /\  ps ) }  <->  A. y ( y  e.  z  <->  E. x
( x  e.  {
x  |  ph }  /\  ps ) ) )
1110exbii 1668 . . . 4  |-  ( E. z  z  =  {
y  |  E. x
( x  e.  {
x  |  ph }  /\  ps ) }  <->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  { x  | 
ph }  /\  ps ) ) )
129, 11mpbir 209 . . 3  |-  E. z 
z  =  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }
1312issetri 3116 . 2  |-  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }  e.  _V
144, 13eqeltrri 2542 1  |-  { y  |  E. x (
ph  /\  ps ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442   _Vcvv 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111
This theorem is referenced by:  zfpair  4693  cshwsexa  12803
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