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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvabdavw | Structured version Visualization version GIF version |
Description: Change bound variable in class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvabdavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvabdavw | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvabdavw.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | cbvsbdavw 36197 | . . 3 ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑦]𝜒)) |
3 | df-clab 2711 | . . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ [𝑡 / 𝑥]𝜓) | |
4 | df-clab 2711 | . . 3 ⊢ (𝑡 ∈ {𝑦 ∣ 𝜒} ↔ [𝑡 / 𝑦]𝜒) | |
5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ 𝑡 ∈ {𝑦 ∣ 𝜒})) |
6 | 5 | eqrdv 2731 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 [wsb 2060 ∈ wcel 2104 {cab 2710 |
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-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 |
This theorem is referenced by: cbvsbcdavw 36200 cbvsbcdavw2 36201 cbvrabdavw 36204 cbviotadavw 36212 cbvixpdavw 36221 cbvrabdavw2 36228 cbvixpdavw2 36237 |
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