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Mirrors > Home > ILE Home > Th. List > cbvopab2 | GIF version |
Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
cbvopab2.1 | ⊢ Ⅎ𝑧𝜑 |
cbvopab2.2 | ⊢ Ⅎ𝑦𝜓 |
cbvopab2.3 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab2 | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
2 | cbvopab2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1457 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 〈𝑥, 𝑧〉 | |
5 | cbvopab2.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
6 | 4, 5 | nfan 1457 | . . . . 5 ⊢ Ⅎ𝑦(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓) |
7 | opeq2 3550 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
8 | 7 | eqeq2d 2051 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑥, 𝑧〉)) |
9 | cbvopab2.3 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | anbi12d 442 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓))) |
11 | 3, 6, 10 | cbvex 1639 | . . . 4 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
12 | 11 | exbii 1496 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
13 | 12 | abbii 2153 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} |
14 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
15 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} | |
16 | 13, 14, 15 | 3eqtr4i 2070 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 {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: cbvoprab3 5580 |
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