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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviotadavw | Structured version Visualization version GIF version |
Description: Change bound variable in a description binder. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbviotadavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbviotadavw | ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviotadavw.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | cbvabdavw 36235 | . . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒}) |
3 | 2 | eqeq1d 2738 | . . . 4 ⊢ (𝜑 → ({𝑥 ∣ 𝜓} = {𝑡} ↔ {𝑦 ∣ 𝜒} = {𝑡})) |
4 | 3 | abbidv 2807 | . . 3 ⊢ (𝜑 → {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}}) |
5 | 4 | unieqd 4918 | . 2 ⊢ (𝜑 → ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}}) |
6 | df-iota 6512 | . 2 ⊢ (℩𝑥𝜓) = ∪ {𝑡 ∣ {𝑥 ∣ 𝜓} = {𝑡}} | |
7 | df-iota 6512 | . 2 ⊢ (℩𝑦𝜒) = ∪ {𝑡 ∣ {𝑦 ∣ 𝜒} = {𝑡}} | |
8 | 5, 6, 7 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 {cab 2713 {csn 4624 ∪ cuni 4905 ℩cio 6510 |
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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-uni 4906 df-iota 6512 |
This theorem is referenced by: cbvriotadavw 36249 cbvriotadavw2 36269 |
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