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| Mirrors > Home > MPE Home > Th. List > cbvcsbv | Structured version Visualization version GIF version | ||
| Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| cbvcsbv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| cbvcsbv | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvcsbv.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 2 | 1 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) | 
| 3 | 2 | cbvsbcvw 3822 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐶) | 
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐶} | 
| 5 | df-csb 3900 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} | |
| 6 | df-csb 3900 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐶} | |
| 7 | 4, 5, 6 | 3eqtr4i 2775 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 [wsbc 3788 ⦋csb 3899 | 
| 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 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 df-csb 3900 | 
| This theorem is referenced by: cbvsumv 15732 cbvprodv 15950 pmatcollpw3lem 22789 precsexlemcbv 28230 cbvprodvw2 36248 poimirlem27 37654 cdleme40v 40471 | 
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