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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvcsbdavw2 | Structured version Visualization version GIF version |
Description: Change bound variable of a proper substitution into a class. General version of cbvcsbdavw 36202. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvcsbdavw2.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
cbvcsbdavw2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cbvcsbdavw2 | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvcsbdavw2.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | cbvcsbdavw2.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) | |
3 | 2 | eleq2d 2823 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
4 | 1, 3 | cbvsbcdavw2 36201 | . . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑡 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑡 ∈ 𝐷)) |
5 | 4 | abbidv 2804 | . 2 ⊢ (𝜑 → {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶} = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷}) |
6 | df-csb 3909 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑡 ∣ [𝐴 / 𝑥]𝑡 ∈ 𝐶} | |
7 | df-csb 3909 | . 2 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑡 ∣ [𝐵 / 𝑦]𝑡 ∈ 𝐷} | |
8 | 5, 6, 7 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {cab 2710 [wsbc 3791 ⦋csb 3908 |
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-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-sbc 3792 df-csb 3909 |
This theorem is referenced by: (None) |
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