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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvsbcvw2 | Structured version Visualization version GIF version |
Description: Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3826. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
cbvsbcvw2.1 | ⊢ 𝐴 = 𝐵 |
cbvsbcvw2.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvsbcvw2 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsbcvw2.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | cbvsbcvw2.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | cbvabv 2808 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
4 | 1, 3 | eleq12i 2830 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∣ 𝜓}) |
5 | df-sbc 3792 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
6 | df-sbc 3792 | . 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝐵 ∈ {𝑦 ∣ 𝜓}) | |
7 | 4, 5, 6 | 3bitr4i 303 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1535 ∈ 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: cbvcsbvw2 36174 |
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