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Mirrors > Home > ILE Home > Th. List > cbvoprab12 | GIF version |
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
cbvoprab12.1 | ⊢ Ⅎ𝑤𝜑 |
cbvoprab12.2 | ⊢ Ⅎ𝑣𝜑 |
cbvoprab12.3 | ⊢ Ⅎ𝑥𝜓 |
cbvoprab12.4 | ⊢ Ⅎ𝑦𝜓 |
cbvoprab12.5 | ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab12 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑤 𝑢 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab12.1 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1544 | . . . 4 ⊢ Ⅎ𝑤(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑣 𝑢 = 〈𝑥, 𝑦〉 | |
5 | cbvoprab12.2 | . . . . 5 ⊢ Ⅎ𝑣𝜑 | |
6 | 4, 5 | nfan 1544 | . . . 4 ⊢ Ⅎ𝑣(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑥 𝑢 = 〈𝑤, 𝑣〉 | |
8 | cbvoprab12.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
9 | 7, 8 | nfan 1544 | . . . 4 ⊢ Ⅎ𝑥(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓) |
10 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑦 𝑢 = 〈𝑤, 𝑣〉 | |
11 | cbvoprab12.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 10, 11 | nfan 1544 | . . . 4 ⊢ Ⅎ𝑦(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓) |
13 | opeq12 3707 | . . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑣〉) | |
14 | 13 | eqeq2d 2151 | . . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝑢 = 〈𝑥, 𝑦〉 ↔ 𝑢 = 〈𝑤, 𝑣〉)) |
15 | cbvoprab12.5 | . . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) | |
16 | 14, 15 | anbi12d 464 | . . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓))) |
17 | 3, 6, 9, 12, 16 | cbvex2 1894 | . . 3 ⊢ (∃𝑥∃𝑦(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑤∃𝑣(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓)) |
18 | 17 | opabbii 3995 | . 2 ⊢ {〈𝑢, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑢, 𝑧〉 ∣ ∃𝑤∃𝑣(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓)} |
19 | dfoprab2 5818 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑢, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
20 | dfoprab2 5818 | . 2 ⊢ {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} = {〈𝑢, 𝑧〉 ∣ ∃𝑤∃𝑣(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓)} | |
21 | 18, 19, 20 | 3eqtr4i 2170 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 〈cop 3530 {copab 3988 {coprab 5775 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-oprab 5778 |
This theorem is referenced by: cbvoprab12v 5846 cbvmpox 5849 dfoprab4f 6091 fmpox 6098 tposoprab 6177 |
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