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Theorem repizf2 4180
Description: Replacement. This version of replacement is stronger than repizf 4134 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 4134 with ax-sep 4136. 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 2755 . . 3 𝑤 ∈ V
21rabex 4162 . 2 {𝑥𝑤 ∣ ∃𝑦𝜑} ∈ V
3 repizf2lem 4179 . . . 4 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
4 nfcv 2332 . . . . . 6 𝑥𝑣
5 nfrab1 2670 . . . . . 6 𝑥{𝑥𝑤 ∣ ∃𝑦𝜑}
64, 5raleqf 2682 . . . . 5 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑))
7 repizf2.1 . . . . . 6 𝑧𝜑
87repizf 4134 . . . . 5 (∀𝑥𝑣 ∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑)
96, 8biimtrrdi 164 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
103, 9biimtrid 152 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
11 df-rab 2477 . . . . . 6 {𝑥𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
12 nfv 1539 . . . . . . . 8 𝑧 𝑥𝑤
137nfex 1648 . . . . . . . 8 𝑧𝑦𝜑
1412, 13nfan 1576 . . . . . . 7 𝑧(𝑥𝑤 ∧ ∃𝑦𝜑)
1514nfab 2337 . . . . . 6 𝑧{𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
1611, 15nfcxfr 2329 . . . . 5 𝑧{𝑥𝑤 ∣ ∃𝑦𝜑}
1716nfeq2 2344 . . . 4 𝑧 𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑}
184, 5raleqf 2682 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣𝑦𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
1917, 18exbid 1627 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∃𝑧𝑥𝑣𝑦𝑧 𝜑 ↔ ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
2010, 19sylibd 149 . 2 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
212, 20vtocle 2826 1 (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wnf 1471  wex 1503  ∃!weu 2038  ∃*wmo 2039  {cab 2175  wral 2468  wrex 2469  {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-coll 4133  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157
This theorem is referenced by: (None)
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