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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvsbcdavw | Structured version Visualization version GIF version |
Description: Change bound variable of a class substitution. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvsbcdavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvsbcdavw | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑦]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsbcdavw.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | cbvabdavw 36199 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒}) |
3 | 2 | eleq2d 2823 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑦 ∣ 𝜒})) |
4 | df-sbc 3792 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
5 | df-sbc 3792 | . 2 ⊢ ([𝐴 / 𝑦]𝜒 ↔ 𝐴 ∈ {𝑦 ∣ 𝜒}) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑦]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2104 {cab 2710 [wsbc 3791 |
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 |
This theorem is referenced by: cbvcsbdavw 36202 |
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