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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvoprab2davw | Structured version Visualization version GIF version |
Description: Change the second bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvoprab2davw.1 | ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvoprab2davw | ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑤〉, 𝑧〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 4881 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑤〉) | |
2 | 1 | adantl 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑤〉) |
3 | 2 | opeq1d 4886 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈〈𝑥, 𝑤〉, 𝑧〉) |
4 | 3 | eqeq2d 2744 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉)) |
5 | cbvoprab2davw.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒)) | |
6 | 4, 5 | anbi12d 631 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → ((𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ (𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒))) |
7 | 6 | exbidv 1917 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒))) |
8 | 7 | cbvexdvaw 2034 | . . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒))) |
9 | 8 | exbidv 1917 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒))) |
10 | 9 | abbidv 2804 | . 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒)}) |
11 | df-oprab 7429 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
12 | df-oprab 7429 | . 2 ⊢ {〈〈𝑥, 𝑤〉, 𝑧〉 ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = 〈〈𝑥, 𝑤〉, 𝑧〉 ∧ 𝜒)} | |
13 | 10, 11, 12 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑤〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∃wex 1774 {cab 2710 〈cop 4636 {coprab 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-oprab 7429 |
This theorem is referenced by: cbvmpo2davw2 36236 |
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