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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 1421 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
2 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 〈𝑧, 𝑦〉 | |
3 | nfs1v 1815 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
4 | 2, 3 | nfan 1457 | . . . . 5 ⊢ Ⅎ𝑥(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
5 | 4 | nfex 1528 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
6 | opeq1 3549 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
7 | 6 | eqeq2d 2051 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
8 | sbequ12 1654 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 7, 8 | anbi12d 442 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
10 | 9 | exbidv 1706 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbvex 1639 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)) |
12 | 11 | abbii 2153 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} |
13 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
14 | df-opab 3819 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} | |
15 | 12, 13, 14 | 3eqtr4i 2070 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∃wex 1381 [wsb 1645 {cab 2026 〈cop 3378 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 |
This theorem is referenced by: (None) |
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