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Theorem repizf2 4018
Description: Replacement. This version of replacement is stronger than repizf 3976 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 3976 with ax-sep 3978. 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 2636 . . 3 𝑤 ∈ V
21rabex 4004 . 2 {𝑥𝑤 ∣ ∃𝑦𝜑} ∈ V
3 repizf2lem 4017 . . . 4 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
4 nfcv 2235 . . . . . 6 𝑥𝑣
5 nfrab1 2560 . . . . . 6 𝑥{𝑥𝑤 ∣ ∃𝑦𝜑}
64, 5raleqf 2572 . . . . 5 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑))
7 repizf2.1 . . . . . 6 𝑧𝜑
87repizf 3976 . . . . 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 2379 . . . . . 6 {𝑥𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
12 nfv 1473 . . . . . . . 8 𝑧 𝑥𝑤
137nfex 1580 . . . . . . . 8 𝑧𝑦𝜑
1412, 13nfan 1509 . . . . . . 7 𝑧(𝑥𝑤 ∧ ∃𝑦𝜑)
1514nfab 2240 . . . . . 6 𝑧{𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
1611, 15nfcxfr 2232 . . . . 5 𝑧{𝑥𝑤 ∣ ∃𝑦𝜑}
1716nfeq2 2247 . . . 4 𝑧 𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑}
184, 5raleqf 2572 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣𝑦𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
1917, 18exbid 1559 . . 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 2707 1 (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wnf 1401  wex 1433  ∃!weu 1955  ∃*wmo 1956  {cab 2081  wral 2370  wrex 2371  {crab 2374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rab 2379  df-v 2635  df-in 3019  df-ss 3026
This theorem is referenced by: (None)
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