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Mirrors > Home > ILE Home > Th. List > cbvoprab3 | GIF version |
Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
cbvoprab3.1 | ⊢ Ⅎ𝑤𝜑 |
cbvoprab3.2 | ⊢ Ⅎ𝑧𝜓 |
cbvoprab3.3 | ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab3 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1462 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab3.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1498 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 1569 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 1569 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | nfv 1462 | . . . . . 6 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
7 | cbvoprab3.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜓 | |
8 | 6, 7 | nfan 1498 | . . . . 5 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
9 | 8 | nfex 1569 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
10 | 9 | nfex 1569 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
11 | cbvoprab3.3 | . . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 11 | anbi2d 452 | . . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
13 | 12 | 2exbidv 1790 | . . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
14 | 5, 10, 13 | cbvopab2 3860 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} |
15 | dfoprab2 5583 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
16 | dfoprab2 5583 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
17 | 14, 15, 16 | 3eqtr4i 2112 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 Ⅎwnf 1390 ∃wex 1422 〈cop 3409 {copab 3846 {coprab 5544 |
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-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-opab 3848 df-oprab 5547 |
This theorem is referenced by: cbvoprab3v 5612 tposoprab 5929 erovlem 6264 |
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