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Theorem repizf2 3962
Description: Replacement. This version of replacement is stronger than repizf 3920 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 3920 with ax-sep 3922. 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 2615 . . 3 𝑤 ∈ V
21rabex 3948 . 2 {𝑥𝑤 ∣ ∃𝑦𝜑} ∈ V
3 repizf2lem 3961 . . . 4 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
4 nfcv 2223 . . . . . 6 𝑥𝑣
5 nfrab1 2539 . . . . . 6 𝑥{𝑥𝑤 ∣ ∃𝑦𝜑}
64, 5raleqf 2551 . . . . 5 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑))
7 repizf2.1 . . . . . 6 𝑧𝜑
87repizf 3920 . . . . 5 (∀𝑥𝑣 ∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑)
96, 8syl6bir 162 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
103, 9syl5bi 150 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
11 df-rab 2362 . . . . . 6 {𝑥𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
12 nfv 1462 . . . . . . . 8 𝑧 𝑥𝑤
137nfex 1569 . . . . . . . 8 𝑧𝑦𝜑
1412, 13nfan 1498 . . . . . . 7 𝑧(𝑥𝑤 ∧ ∃𝑦𝜑)
1514nfab 2227 . . . . . 6 𝑧{𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
1611, 15nfcxfr 2220 . . . . 5 𝑧{𝑥𝑤 ∣ ∃𝑦𝜑}
1716nfeq2 2234 . . . 4 𝑧 𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑}
184, 5raleqf 2551 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣𝑦𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
1917, 18exbid 1548 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∃𝑧𝑥𝑣𝑦𝑧 𝜑 ↔ ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
2010, 19sylibd 147 . 2 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
212, 20vtocle 2683 1 (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wnf 1390  wex 1422  ∃!weu 1943  ∃*wmo 1944  {cab 2069  wral 2353  wrex 2354  {crab 2357
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2614  df-in 2990  df-ss 2997
This theorem is referenced by: (None)
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