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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvopab1davw | Structured version Visualization version GIF version |
Description: Change the first bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvopab1davw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvopab1davw | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑧, 𝑦〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4880 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
2 | 1 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) |
3 | 2 | eqeq2d 2744 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑡 = 〈𝑥, 𝑦〉 ↔ 𝑡 = 〈𝑧, 𝑦〉)) |
4 | cbvopab1davw.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | anbi12d 631 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → ((𝑡 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝑡 = 〈𝑧, 𝑦〉 ∧ 𝜒))) |
6 | 5 | exbidv 1917 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑦(𝑡 = 〈𝑧, 𝑦〉 ∧ 𝜒))) |
7 | 6 | cbvexdvaw 2034 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑧∃𝑦(𝑡 = 〈𝑧, 𝑦〉 ∧ 𝜒))) |
8 | 7 | abbidv 2804 | . 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ 𝜓)} = {𝑡 ∣ ∃𝑧∃𝑦(𝑡 = 〈𝑧, 𝑦〉 ∧ 𝜒)}) |
9 | df-opab 5212 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
10 | df-opab 5212 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ 𝜒} = {𝑡 ∣ ∃𝑧∃𝑦(𝑡 = 〈𝑧, 𝑦〉 ∧ 𝜒)} | |
11 | 8, 9, 10 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑧, 𝑦〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∃wex 1774 {cab 2710 〈cop 4636 {copab 5211 |
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-opab 5212 |
This theorem is referenced by: cbvmptdavw 36210 cbvmptdavw2 36231 |
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