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Theorem repizf2 3989
Description: Replacement. This version of replacement is stronger than repizf 3947 in the sense that  ph does not need to map all values of  x in  w to a value of  y. The resulting set contains those elements for which there is a value of  y and in that sense, this theorem combines repizf 3947 with ax-sep 3949. Another variation would be  A. x  e.  w E* y ph  ->  { y  |  E. x ( x  e.  w  /\  ph ) }  e.  _V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
Hypothesis
Ref Expression
repizf2.1  |-  F/ z
ph
Assertion
Ref Expression
repizf2  |-  ( A. x  e.  w  E* y ph  ->  E. z A. x  e.  { x  e.  w  |  E. y ph } E. y  e.  z  ph )
Distinct variable group:    x, y, z, w
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem repizf2
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 vex 2622 . . 3  |-  w  e. 
_V
21rabex 3975 . 2  |-  { x  e.  w  |  E. y ph }  e.  _V
3 repizf2lem 3988 . . . 4  |-  ( A. x  e.  w  E* y ph  <->  A. x  e.  {
x  e.  w  |  E. y ph } E! y ph )
4 nfcv 2228 . . . . . 6  |-  F/_ x
v
5 nfrab1 2546 . . . . . 6  |-  F/_ x { x  e.  w  |  E. y ph }
64, 5raleqf 2558 . . . . 5  |-  ( v  =  { x  e.  w  |  E. y ph }  ->  ( A. x  e.  v  E! y ph  <->  A. x  e.  {
x  e.  w  |  E. y ph } E! y ph ) )
7 repizf2.1 . . . . . 6  |-  F/ z
ph
87repizf 3947 . . . . 5  |-  ( A. x  e.  v  E! y ph  ->  E. z A. x  e.  v  E. y  e.  z  ph )
96, 8syl6bir 162 . . . 4  |-  ( v  =  { x  e.  w  |  E. y ph }  ->  ( A. x  e.  { x  e.  w  |  E. y ph } E! y
ph  ->  E. z A. x  e.  v  E. y  e.  z  ph ) )
103, 9syl5bi 150 . . 3  |-  ( v  =  { x  e.  w  |  E. y ph }  ->  ( A. x  e.  w  E* y ph  ->  E. z A. x  e.  v  E. y  e.  z  ph ) )
11 df-rab 2368 . . . . . 6  |-  { x  e.  w  |  E. y ph }  =  {
x  |  ( x  e.  w  /\  E. y ph ) }
12 nfv 1466 . . . . . . . 8  |-  F/ z  x  e.  w
137nfex 1573 . . . . . . . 8  |-  F/ z E. y ph
1412, 13nfan 1502 . . . . . . 7  |-  F/ z ( x  e.  w  /\  E. y ph )
1514nfab 2233 . . . . . 6  |-  F/_ z { x  |  (
x  e.  w  /\  E. y ph ) }
1611, 15nfcxfr 2225 . . . . 5  |-  F/_ z { x  e.  w  |  E. y ph }
1716nfeq2 2240 . . . 4  |-  F/ z  v  =  { x  e.  w  |  E. y ph }
184, 5raleqf 2558 . . . 4  |-  ( v  =  { x  e.  w  |  E. y ph }  ->  ( A. x  e.  v  E. y  e.  z  ph  <->  A. x  e.  { x  e.  w  |  E. y ph } E. y  e.  z  ph ) )
1917, 18exbid 1552 . . 3  |-  ( v  =  { x  e.  w  |  E. y ph }  ->  ( E. z A. x  e.  v  E. y  e.  z 
ph 
<->  E. z A. x  e.  { x  e.  w  |  E. y ph } E. y  e.  z  ph ) )
2010, 19sylibd 147 . 2  |-  ( v  =  { x  e.  w  |  E. y ph }  ->  ( A. x  e.  w  E* y ph  ->  E. z A. x  e.  { x  e.  w  |  E. y ph } E. y  e.  z  ph ) )
212, 20vtocle 2693 1  |-  ( A. x  e.  w  E* y ph  ->  E. z A. x  e.  { x  e.  w  |  E. y ph } E. y  e.  z  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   F/wnf 1394   E.wex 1426   E!weu 1948   E*wmo 1949   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-in 3003  df-ss 3010
This theorem is referenced by: (None)
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