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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 3821. (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 2811 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
| 4 | 1, 3 | eleq12i 2833 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∣ 𝜓}) |
| 5 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
| 6 | df-sbc 3788 | . 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 1540 ∈ wcel 2108 {cab 2713 [wsbc 3787 |
| 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-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 |
| This theorem is referenced by: cbvcsbvw2 36210 |
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