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Theorem repizf2 4092
Description: Replacement. This version of replacement is stronger than repizf 4050 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 4050 with ax-sep 4052. Another variation would be 𝑥𝑤∃*𝑦𝜑 → {𝑦 ∣ ∃𝑥(𝑥𝑤𝜑)} ∈ 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 (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 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 2692 . . 3 𝑤 ∈ V
21rabex 4078 . 2 {𝑥𝑤 ∣ ∃𝑦𝜑} ∈ V
3 repizf2lem 4091 . . . 4 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
4 nfcv 2282 . . . . . 6 𝑥𝑣
5 nfrab1 2613 . . . . . 6 𝑥{𝑥𝑤 ∣ ∃𝑦𝜑}
64, 5raleqf 2625 . . . . 5 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑))
7 repizf2.1 . . . . . 6 𝑧𝜑
87repizf 4050 . . . . 5 (∀𝑥𝑣 ∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑)
96, 8syl6bir 163 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
103, 9syl5bi 151 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
11 df-rab 2426 . . . . . 6 {𝑥𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
12 nfv 1509 . . . . . . . 8 𝑧 𝑥𝑤
137nfex 1617 . . . . . . . 8 𝑧𝑦𝜑
1412, 13nfan 1545 . . . . . . 7 𝑧(𝑥𝑤 ∧ ∃𝑦𝜑)
1514nfab 2287 . . . . . 6 𝑧{𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
1611, 15nfcxfr 2279 . . . . 5 𝑧{𝑥𝑤 ∣ ∃𝑦𝜑}
1716nfeq2 2294 . . . 4 𝑧 𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑}
184, 5raleqf 2625 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣𝑦𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
1917, 18exbid 1596 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∃𝑧𝑥𝑣𝑦𝑧 𝜑 ↔ ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
2010, 19sylibd 148 . 2 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
212, 20vtocle 2763 1 (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wnf 1437  wex 1469  ∃!weu 2000  ∃*wmo 2001  {cab 2126  wral 2417  wrex 2418  {crab 2421
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-in 3080  df-ss 3087
This theorem is referenced by: (None)
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