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Mirrors > Home > ILE Home > Th. List > cbvopab | GIF version |
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
cbvopab.1 | ⊢ Ⅎ𝑧𝜑 |
cbvopab.2 | ⊢ Ⅎ𝑤𝜑 |
cbvopab.3 | ⊢ Ⅎ𝑥𝜓 |
cbvopab.4 | ⊢ Ⅎ𝑦𝜓 |
cbvopab.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1473 | . . . . 5 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvopab.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1509 | . . . 4 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1473 | . . . . 5 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
5 | cbvopab.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
6 | 4, 5 | nfan 1509 | . . . 4 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | nfv 1473 | . . . . 5 ⊢ Ⅎ𝑥 𝑣 = 〈𝑧, 𝑤〉 | |
8 | cbvopab.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
9 | 7, 8 | nfan 1509 | . . . 4 ⊢ Ⅎ𝑥(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓) |
10 | nfv 1473 | . . . . 5 ⊢ Ⅎ𝑦 𝑣 = 〈𝑧, 𝑤〉 | |
11 | cbvopab.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 10, 11 | nfan 1509 | . . . 4 ⊢ Ⅎ𝑦(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓) |
13 | opeq12 3646 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
14 | 13 | eqeq2d 2106 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 〈𝑥, 𝑦〉 ↔ 𝑣 = 〈𝑧, 𝑤〉)) |
15 | cbvopab.5 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
16 | 14, 15 | anbi12d 458 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓))) |
17 | 3, 6, 9, 12, 16 | cbvex2 1852 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)) |
18 | 17 | abbii 2210 | . 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} |
19 | df-opab 3922 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
20 | df-opab 3922 | . 2 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} | |
21 | 18, 19, 20 | 3eqtr4i 2125 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 Ⅎwnf 1401 ∃wex 1433 {cab 2081 〈cop 3469 {copab 3920 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 df-op 3475 df-opab 3922 |
This theorem is referenced by: cbvopabv 3932 opelopabsb 4111 |
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