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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 1539 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab1.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1576 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 1648 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | nfv 1539 | . . . . . 6 ⊢ Ⅎ𝑥 𝑣 = 〈𝑤, 𝑦〉 | |
6 | cbvoprab1.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
7 | 5, 6 | nfan 1576 | . . . . 5 ⊢ Ⅎ𝑥(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
8 | 7 | nfex 1648 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
9 | opeq1 3793 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑦〉) | |
10 | 9 | eqeq2d 2201 | . . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = 〈𝑥, 𝑦〉 ↔ 𝑣 = 〈𝑤, 𝑦〉)) |
11 | cbvoprab1.3 | . . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | anbi12d 473 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
13 | 12 | exbidv 1836 | . . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
14 | 4, 8, 13 | cbvex 1767 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)) |
15 | 14 | opabbii 4085 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} |
16 | dfoprab2 5943 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
17 | dfoprab2 5943 | . 2 ⊢ {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} | |
18 | 15, 16, 17 | 3eqtr4i 2220 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 Ⅎwnf 1471 ∃wex 1503 〈cop 3610 {copab 4078 {coprab 5897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-oprab 5900 |
This theorem is referenced by: (None) |
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