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Mirrors > Home > ILE Home > Th. List > cbvopab1s | GIF version |
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.) |
Ref | Expression |
---|---|
cbvopab1s | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1462 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
2 | nfv 1462 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 〈𝑧, 𝑦〉 | |
3 | nfs1v 1858 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
4 | 2, 3 | nfan 1498 | . . . . 5 ⊢ Ⅎ𝑥(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
5 | 4 | nfex 1569 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
6 | opeq1 3590 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
7 | 6 | eqeq2d 2094 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
8 | sbequ12 1696 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 7, 8 | anbi12d 457 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
10 | 9 | exbidv 1748 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbvex 1681 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)) |
12 | 11 | abbii 2198 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} |
13 | df-opab 3860 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
14 | df-opab 3860 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} | |
15 | 12, 13, 14 | 3eqtr4i 2113 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1285 ∃wex 1422 [wsb 1687 {cab 2069 〈cop 3419 {copab 3858 |
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-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2612 df-un 2986 df-sn 3422 df-pr 3423 df-op 3425 df-opab 3860 |
This theorem is referenced by: (None) |
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