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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvoprab2vw | Structured version Visualization version GIF version | ||
| Description: Change the second bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| cbvoprab2vw.1 | ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| cbvoprab2vw | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑤〉, 𝑧〉 ∣ 𝜒} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opeq2 4872 | . . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑤〉) | |
| 2 | 1 | opeq1d 4877 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈〈𝑥, 𝑤〉, 𝑧〉) |
| 3 | 2 | eqeq2d 2747 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉)) |
| 4 | cbvoprab2vw.1 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒)) | |
| 5 | 3, 4 | anbi12d 632 | . . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ (𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒))) |
| 6 | 5 | exbidv 1921 | . . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒))) |
| 7 | 6 | cbvexvw 2036 | . . . 4 ⊢ (∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒)) |
| 8 | 7 | exbii 1848 | . . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒)) |
| 9 | 8 | abbii 2808 | . 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒)} |
| 10 | df-oprab 7433 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
| 11 | df-oprab 7433 | . 2 ⊢ {〈〈𝑥, 𝑤〉, 𝑧〉 ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒)} | |
| 12 | 9, 10, 11 | 3eqtr4i 2774 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑤〉, 𝑧〉 ∣ 𝜒} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 {cab 2713 〈cop 4630 {coprab 7430 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-oprab 7433 |
| This theorem is referenced by: cbvmpo2vw2 36223 |
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