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Theorem zfrep4 4546
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 2416 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
21anbi1i 699 . . . 4  |-  ( ( x  e.  { x  |  ph }  /\  ps ) 
<->  ( ph  /\  ps ) )
32exbii 1714 . . 3  |-  ( E. x ( x  e. 
{ x  |  ph }  /\  ps )  <->  E. x
( ph  /\  ps )
)
43abbii 2563 . 2  |-  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }  =  {
y  |  E. x
( ph  /\  ps ) }
5 nfab1 2593 . . . . 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 198 . . . . 5  |-  ( x  e.  { x  | 
ph }  ->  E. z A. y ( ps  ->  y  =  z ) )
95, 6, 8zfrepclf 4544 . . . 4  |-  E. z A. y ( y  e.  z  <->  E. x ( x  e.  { x  | 
ph }  /\  ps ) )
10 abeq2 2553 . . . . 5  |-  ( z  =  { y  |  E. x ( x  e.  { x  | 
ph }  /\  ps ) }  <->  A. y ( y  e.  z  <->  E. x
( x  e.  {
x  |  ph }  /\  ps ) ) )
1110exbii 1714 . . . 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 212 . . 3  |-  E. z 
z  =  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }
1312issetri 3094 . 2  |-  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }  e.  _V
144, 13eqeltrri 2514 1  |-  { y  |  E. x (
ph  /\  ps ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   _Vcvv 3087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089
This theorem is referenced by:  zfpair  4659  cshwsexa  12908
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