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Mirrors > Home > ILE Home > Th. List > repizf2 | GIF version |
Description: Replacement. This version of replacement is stronger than repizf 3864 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 3864 with ax-sep 3866. Another variation would be ∀x ∈ w∃*yφ → {y ∣ ∃x(x ∈ w ∧ φ)} ∈ V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.) |
Ref | Expression |
---|---|
repizf2.1 | ⊢ Ⅎzφ |
Ref | Expression |
---|---|
repizf2 | ⊢ (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . 3 ⊢ w ∈ V | |
2 | 1 | rabex 3892 | . 2 ⊢ {x ∈ w ∣ ∃yφ} ∈ V |
3 | repizf2lem 3905 | . . . 4 ⊢ (∀x ∈ w ∃*yφ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ) | |
4 | nfcv 2175 | . . . . . 6 ⊢ Ⅎxv | |
5 | nfrab1 2483 | . . . . . 6 ⊢ Ⅎx{x ∈ w ∣ ∃yφ} | |
6 | 4, 5 | raleqf 2495 | . . . . 5 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ v ∃!yφ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ)) |
7 | repizf2.1 | . . . . . 6 ⊢ Ⅎzφ | |
8 | 7 | repizf 3864 | . . . . 5 ⊢ (∀x ∈ v ∃!yφ → ∃z∀x ∈ v ∃y ∈ z φ) |
9 | 6, 8 | syl6bir 153 | . . . 4 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ {x ∈ w ∣ ∃yφ}∃!yφ → ∃z∀x ∈ v ∃y ∈ z φ)) |
10 | 3, 9 | syl5bi 141 | . . 3 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ w ∃*yφ → ∃z∀x ∈ v ∃y ∈ z φ)) |
11 | df-rab 2309 | . . . . . 6 ⊢ {x ∈ w ∣ ∃yφ} = {x ∣ (x ∈ w ∧ ∃yφ)} | |
12 | nfv 1418 | . . . . . . . 8 ⊢ Ⅎz x ∈ w | |
13 | 7 | nfex 1525 | . . . . . . . 8 ⊢ Ⅎz∃yφ |
14 | 12, 13 | nfan 1454 | . . . . . . 7 ⊢ Ⅎz(x ∈ w ∧ ∃yφ) |
15 | 14 | nfab 2179 | . . . . . 6 ⊢ Ⅎz{x ∣ (x ∈ w ∧ ∃yφ)} |
16 | 11, 15 | nfcxfr 2172 | . . . . 5 ⊢ Ⅎz{x ∈ w ∣ ∃yφ} |
17 | 16 | nfeq2 2186 | . . . 4 ⊢ Ⅎz v = {x ∈ w ∣ ∃yφ} |
18 | 4, 5 | raleqf 2495 | . . . 4 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ v ∃y ∈ z φ ↔ ∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ)) |
19 | 17, 18 | exbid 1504 | . . 3 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∃z∀x ∈ v ∃y ∈ z φ ↔ ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ)) |
20 | 10, 19 | sylibd 138 | . 2 ⊢ (v = {x ∈ w ∣ ∃yφ} → (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ)) |
21 | 2, 20 | vtocle 2621 | 1 ⊢ (∀x ∈ w ∃*yφ → ∃z∀x ∈ {x ∈ w ∣ ∃yφ}∃y ∈ z φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 Ⅎwnf 1346 ∃wex 1378 ∃!weu 1897 ∃*wmo 1898 {cab 2023 ∀wral 2300 ∃wrex 2301 {crab 2304 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rab 2309 df-v 2553 df-in 2918 df-ss 2925 |
This theorem is referenced by: (None) |
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