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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 1473 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
2 | nfv 1473 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 〈𝑧, 𝑦〉 | |
3 | nfs1v 1870 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
4 | 2, 3 | nfan 1509 | . . . . 5 ⊢ Ⅎ𝑥(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
5 | 4 | nfex 1580 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
6 | opeq1 3644 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
7 | 6 | eqeq2d 2106 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
8 | sbequ12 1708 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 7, 8 | anbi12d 458 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
10 | 9 | exbidv 1760 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbvex 1693 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)) |
12 | 11 | abbii 2210 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} |
13 | df-opab 3922 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
14 | df-opab 3922 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} | |
15 | 12, 13, 14 | 3eqtr4i 2125 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1296 ∃wex 1433 [wsb 1699 {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: (None) |
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