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Theorem repizf2 4081
Description: Replacement. This version of replacement is stronger than repizf 4039 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 4039 with ax-sep 4041. 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 2684 . . 3  |-  w  e. 
_V
21rabex 4067 . 2  |-  { x  e.  w  |  E. y ph }  e.  _V
3 repizf2lem 4080 . . . 4  |-  ( A. x  e.  w  E* y ph  <->  A. x  e.  {
x  e.  w  |  E. y ph } E! y ph )
4 nfcv 2279 . . . . . 6  |-  F/_ x
v
5 nfrab1 2608 . . . . . 6  |-  F/_ x { x  e.  w  |  E. y ph }
64, 5raleqf 2620 . . . . 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 4039 . . . . 5  |-  ( A. x  e.  v  E! y ph  ->  E. z A. x  e.  v  E. y  e.  z  ph )
96, 8syl6bir 163 . . . 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 151 . . 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 2423 . . . . . 6  |-  { x  e.  w  |  E. y ph }  =  {
x  |  ( x  e.  w  /\  E. y ph ) }
12 nfv 1508 . . . . . . . 8  |-  F/ z  x  e.  w
137nfex 1616 . . . . . . . 8  |-  F/ z E. y ph
1412, 13nfan 1544 . . . . . . 7  |-  F/ z ( x  e.  w  /\  E. y ph )
1514nfab 2284 . . . . . 6  |-  F/_ z { x  |  (
x  e.  w  /\  E. y ph ) }
1611, 15nfcxfr 2276 . . . . 5  |-  F/_ z { x  e.  w  |  E. y ph }
1716nfeq2 2291 . . . 4  |-  F/ z  v  =  { x  e.  w  |  E. y ph }
184, 5raleqf 2620 . . . 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 1595 . . 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 148 . 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 2755 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 103    = wceq 1331   F/wnf 1436   E.wex 1468   E!weu 1997   E*wmo 1998   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-in 3072  df-ss 3079
This theorem is referenced by: (None)
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