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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 1515 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab1.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1552 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 1624 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | nfv 1515 | . . . . . 6 ⊢ Ⅎ𝑥 𝑣 = 〈𝑤, 𝑦〉 | |
6 | cbvoprab1.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
7 | 5, 6 | nfan 1552 | . . . . 5 ⊢ Ⅎ𝑥(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
8 | 7 | nfex 1624 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
9 | opeq1 3753 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑦〉) | |
10 | 9 | eqeq2d 2176 | . . . . . 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 1812 | . . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
14 | 4, 8, 13 | cbvex 1743 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)) |
15 | 14 | opabbii 4044 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} |
16 | dfoprab2 5881 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
17 | dfoprab2 5881 | . 2 ⊢ {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} | |
18 | 15, 16, 17 | 3eqtr4i 2195 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 Ⅎwnf 1447 ∃wex 1479 〈cop 3574 {copab 4037 {coprab 5838 |
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-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
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 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-opab 4039 df-oprab 5841 |
This theorem is referenced by: (None) |
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