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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 1515 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
2 | cbvopab2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1552 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1515 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 〈𝑥, 𝑧〉 | |
5 | cbvopab2.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
6 | 4, 5 | nfan 1552 | . . . . 5 ⊢ Ⅎ𝑦(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓) |
7 | opeq2 3756 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
8 | 7 | eqeq2d 2176 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑥, 𝑧〉)) |
9 | cbvopab2.3 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | anbi12d 465 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓))) |
11 | 3, 6, 10 | cbvex 1743 | . . . 4 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
12 | 11 | exbii 1592 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
13 | 12 | abbii 2280 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} |
14 | df-opab 4041 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
15 | df-opab 4041 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} | |
16 | 13, 14, 15 | 3eqtr4i 2195 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 Ⅎwnf 1447 ∃wex 1479 {cab 2150 〈cop 3576 {copab 4039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2726 df-un 3118 df-sn 3579 df-pr 3580 df-op 3582 df-opab 4041 |
This theorem is referenced by: cbvoprab3 5912 |
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