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Theorem repizf2 3942
Description: Replacement. This version of replacement is stronger than repizf 3900 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 3900 with ax-sep 3902. 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 2577 . . 3 𝑤 ∈ V
21rabex 3928 . 2 {𝑥𝑤 ∣ ∃𝑦𝜑} ∈ V
3 repizf2lem 3941 . . . 4 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
4 nfcv 2194 . . . . . 6 𝑥𝑣
5 nfrab1 2506 . . . . . 6 𝑥{𝑥𝑤 ∣ ∃𝑦𝜑}
64, 5raleqf 2518 . . . . 5 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑))
7 repizf2.1 . . . . . 6 𝑧𝜑
87repizf 3900 . . . . 5 (∀𝑥𝑣 ∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑)
96, 8syl6bir 157 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
103, 9syl5bi 145 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥𝑣𝑦𝑧 𝜑))
11 df-rab 2332 . . . . . 6 {𝑥𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
12 nfv 1437 . . . . . . . 8 𝑧 𝑥𝑤
137nfex 1544 . . . . . . . 8 𝑧𝑦𝜑
1412, 13nfan 1473 . . . . . . 7 𝑧(𝑥𝑤 ∧ ∃𝑦𝜑)
1514nfab 2198 . . . . . 6 𝑧{𝑥 ∣ (𝑥𝑤 ∧ ∃𝑦𝜑)}
1611, 15nfcxfr 2191 . . . . 5 𝑧{𝑥𝑤 ∣ ∃𝑦𝜑}
1716nfeq2 2205 . . . 4 𝑧 𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑}
184, 5raleqf 2518 . . . 4 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑣𝑦𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
1917, 18exbid 1523 . . 3 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∃𝑧𝑥𝑣𝑦𝑧 𝜑 ↔ ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
2010, 19sylibd 142 . 2 (𝑣 = {𝑥𝑤 ∣ ∃𝑦𝜑} → (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑))
212, 20vtocle 2644 1 (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wnf 1365  wex 1397  ∃!weu 1916  ∃*wmo 1917  {cab 2042  wral 2323  wrex 2324  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
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