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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 1538 | . . . . 5 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvopab.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1575 | . . . 4 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1538 | . . . . 5 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
5 | cbvopab.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
6 | 4, 5 | nfan 1575 | . . . 4 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | nfv 1538 | . . . . 5 ⊢ Ⅎ𝑥 𝑣 = 〈𝑧, 𝑤〉 | |
8 | cbvopab.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
9 | 7, 8 | nfan 1575 | . . . 4 ⊢ Ⅎ𝑥(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓) |
10 | nfv 1538 | . . . . 5 ⊢ Ⅎ𝑦 𝑣 = 〈𝑧, 𝑤〉 | |
11 | cbvopab.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 10, 11 | nfan 1575 | . . . 4 ⊢ Ⅎ𝑦(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓) |
13 | opeq12 3792 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
14 | 13 | eqeq2d 2199 | . . . . 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 473 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓))) |
17 | 3, 6, 9, 12, 16 | cbvex2 1932 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)) |
18 | 17 | abbii 2303 | . 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} |
19 | df-opab 4077 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
20 | df-opab 4077 | . 2 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} | |
21 | 18, 19, 20 | 3eqtr4i 2218 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 Ⅎwnf 1470 ∃wex 1502 {cab 2173 〈cop 3607 {copab 4075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 |
This theorem is referenced by: cbvopabv 4087 opelopabsb 4272 |
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