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Theorem repizf2 3905
Description: Replacement. This version of replacement is stronger than repizf 3863 in the sense that φ 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 3863 with ax-sep 3865. Another variation would be x w∃*yφ → {yx(x w φ)} V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
Hypothesis
Ref Expression
repizf2.1 zφ
Assertion
Ref Expression
repizf2 (x w ∃*yφzx {x wyφ}y z φ)
Distinct variable group:   x,y,z,w
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem repizf2
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . 3 w V
21rabex 3891 . 2 {x wyφ} V
3 repizf2lem 3904 . . . 4 (x w ∃*yφx {x wyφ}∃!yφ)
4 nfcv 2175 . . . . . 6 xv
5 nfrab1 2483 . . . . . 6 x{x wyφ}
64, 5raleqf 2495 . . . . 5 (v = {x wyφ} → (x v ∃!yφx {x wyφ}∃!yφ))
7 repizf2.1 . . . . . 6 zφ
87repizf 3863 . . . . 5 (x v ∃!yφzx v y z φ)
96, 8syl6bir 153 . . . 4 (v = {x wyφ} → (x {x wyφ}∃!yφzx v y z φ))
103, 9syl5bi 141 . . 3 (v = {x wyφ} → (x w ∃*yφzx v y z φ))
11 df-rab 2309 . . . . . 6 {x wyφ} = {x ∣ (x w yφ)}
12 nfv 1418 . . . . . . . 8 z x w
137nfex 1525 . . . . . . . 8 zyφ
1412, 13nfan 1454 . . . . . . 7 z(x w yφ)
1514nfab 2179 . . . . . 6 z{x ∣ (x w yφ)}
1611, 15nfcxfr 2172 . . . . 5 z{x wyφ}
1716nfeq2 2186 . . . 4 z v = {x wyφ}
184, 5raleqf 2495 . . . 4 (v = {x wyφ} → (x v y z φx {x wyφ}y z φ))
1917, 18exbid 1504 . . 3 (v = {x wyφ} → (zx v y z φzx {x wyφ}y z φ))
2010, 19sylibd 138 . 2 (v = {x wyφ} → (x w ∃*yφzx {x wyφ}y z φ))
212, 20vtocle 2621 1 (x w ∃*yφzx {x wyφ}y z φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wnf 1346  wex 1378  ∃!weu 1897  ∃*wmo 1898  {cab 2023  wral 2300  wrex 2301  {crab 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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