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| Mirrors > Home > MPE Home > Th. List > cbvsbcvw | Structured version Visualization version GIF version | ||
| Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3823 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2376. (Revised by GG, 10-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| cbvsbcvw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvsbcvw | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvsbcvw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | cbvabv 2811 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | 
| 3 | 2 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | 
| 4 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
| 5 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 {cab 2713 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: cbvcsbv 3910 frpoins3xpg 8166 frpoins3xp3g 8167 fpwwe2cbv 10671 reuf1odnf 47124 | 
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