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Theorem zfrep4 4522
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 2441 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
21anbi1i 695 . . . 4  |-  ( ( x  e.  { x  |  ph }  /\  ps ) 
<->  ( ph  /\  ps ) )
32exbii 1635 . . 3  |-  ( E. x ( x  e. 
{ x  |  ph }  /\  ps )  <->  E. x
( ph  /\  ps )
)
43abbii 2588 . 2  |-  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }  =  {
y  |  E. x
( ph  /\  ps ) }
5 nfab1 2618 . . . . 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 4520 . . . 4  |-  E. z A. y ( y  e.  z  <->  E. x ( x  e.  { x  | 
ph }  /\  ps ) )
10 abeq2 2578 . . . . 5  |-  ( z  =  { y  |  E. x ( x  e.  { x  | 
ph }  /\  ps ) }  <->  A. y ( y  e.  z  <->  E. x
( x  e.  {
x  |  ph }  /\  ps ) ) )
1110exbii 1635 . . . 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 3085 . 2  |-  { y  |  E. x ( x  e.  { x  |  ph }  /\  ps ) }  e.  _V
144, 13eqeltrri 2539 1  |-  { y  |  E. x (
ph  /\  ps ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439   _Vcvv 3078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080
This theorem is referenced by:  zfpair  4640  cshwsexa  12580
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