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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvrabdavw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvrabdavw2.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
cbvrabdavw2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvrabdavw2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2823 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | 1 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
3 | cbvrabdavw2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
4 | 3 | eleq2d 2826 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
5 | 2, 4 | bitrd 279 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
6 | cbvrabdavw2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
7 | 5, 6 | anbi12d 632 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒))) |
8 | 7 | cbvabdavw 36235 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}) |
9 | df-rab 3436 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
10 | df-rab 3436 | . 2 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)} | |
11 | 8, 9, 10 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2713 {crab 3435 |
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-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 |
This theorem is referenced by: (None) |
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