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Mirrors > Home > ILE Home > Th. List > cbvoprab1 | GIF version |
Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
cbvoprab1.1 | ⊢ Ⅎ𝑤𝜑 |
cbvoprab1.2 | ⊢ Ⅎ𝑥𝜓 |
cbvoprab1.3 | ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab1 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab1.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1545 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 1617 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | nfv 1509 | . . . . . 6 ⊢ Ⅎ𝑥 𝑣 = 〈𝑤, 𝑦〉 | |
6 | cbvoprab1.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
7 | 5, 6 | nfan 1545 | . . . . 5 ⊢ Ⅎ𝑥(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
8 | 7 | nfex 1617 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
9 | opeq1 3713 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑦〉) | |
10 | 9 | eqeq2d 2152 | . . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = 〈𝑥, 𝑦〉 ↔ 𝑣 = 〈𝑤, 𝑦〉)) |
11 | cbvoprab1.3 | . . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | anbi12d 465 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
13 | 12 | exbidv 1798 | . . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
14 | 4, 8, 13 | cbvex 1730 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)) |
15 | 14 | opabbii 4003 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} |
16 | dfoprab2 5826 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
17 | dfoprab2 5826 | . 2 ⊢ {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} | |
18 | 15, 16, 17 | 3eqtr4i 2171 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 Ⅎwnf 1437 ∃wex 1469 〈cop 3535 {copab 3996 {coprab 5783 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-oprab 5786 |
This theorem is referenced by: (None) |
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