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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 1493 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab3.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1529 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 1601 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 1601 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | nfv 1493 | . . . . . 6 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
7 | cbvoprab3.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜓 | |
8 | 6, 7 | nfan 1529 | . . . . 5 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
9 | 8 | nfex 1601 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
10 | 9 | nfex 1601 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
11 | cbvoprab3.3 | . . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 11 | anbi2d 459 | . . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
13 | 12 | 2exbidv 1824 | . . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
14 | 5, 10, 13 | cbvopab2 3972 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} |
15 | dfoprab2 5786 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
16 | dfoprab2 5786 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
17 | 14, 15, 16 | 3eqtr4i 2148 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 Ⅎwnf 1421 ∃wex 1453 〈cop 3500 {copab 3958 {coprab 5743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-oprab 5746 |
This theorem is referenced by: cbvoprab3v 5816 tposoprab 6145 erovlem 6489 |
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