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Theorem repizf2 3943
Description: Replacement. This version of replacement is stronger than repizf 3901 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 3901 with ax-sep 3903. 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 2577 . . 3  |-  w  e. 
_V
21rabex 3929 . 2  |-  { x  e.  w  |  E. y ph }  e.  _V
3 repizf2lem 3942 . . . 4  |-  ( A. x  e.  w  E* y ph  <->  A. x  e.  {
x  e.  w  |  E. y ph } E! y ph )
4 nfcv 2194 . . . . . 6  |-  F/_ x
v
5 nfrab1 2506 . . . . . 6  |-  F/_ x { x  e.  w  |  E. y ph }
64, 5raleqf 2518 . . . . 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 3901 . . . . 5  |-  ( A. x  e.  v  E! y ph  ->  E. z A. x  e.  v  E. y  e.  z  ph )
96, 8syl6bir 157 . . . 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 145 . . 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 2332 . . . . . 6  |-  { x  e.  w  |  E. y ph }  =  {
x  |  ( x  e.  w  /\  E. y ph ) }
12 nfv 1437 . . . . . . . 8  |-  F/ z  x  e.  w
137nfex 1544 . . . . . . . 8  |-  F/ z E. y ph
1412, 13nfan 1473 . . . . . . 7  |-  F/ z ( x  e.  w  /\  E. y ph )
1514nfab 2198 . . . . . 6  |-  F/_ z { x  |  (
x  e.  w  /\  E. y ph ) }
1611, 15nfcxfr 2191 . . . . 5  |-  F/_ z { x  e.  w  |  E. y ph }
1716nfeq2 2205 . . . 4  |-  F/ z  v  =  { x  e.  w  |  E. y ph }
184, 5raleqf 2518 . . . 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 1523 . . 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 142 . 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 2644 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 101    = wceq 1259   F/wnf 1365   E.wex 1397   E!weu 1916   E*wmo 1917   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-in 2952  df-ss 2959
This theorem is referenced by: (None)
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