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| Mirrors > Home > MPE Home > Th. List > cbvcsbw | Structured version Visualization version GIF version | ||
| Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Version of cbvcsb 3910 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Jeff Hankins, 13-Sep-2009.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) |
| Ref | Expression |
|---|---|
| cbvcsbw.1 | ⊢ Ⅎ𝑦𝐶 |
| cbvcsbw.2 | ⊢ Ⅎ𝑥𝐷 |
| cbvcsbw.3 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| cbvcsbw | ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvcsbw.1 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
| 2 | 1 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 |
| 3 | cbvcsbw.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
| 4 | 3 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 |
| 5 | cbvcsbw.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
| 6 | 5 | eleq2d 2827 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
| 7 | 2, 4, 6 | cbvsbcw 3821 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) |
| 8 | 7 | abbii 2809 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} |
| 9 | df-csb 3900 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
| 10 | df-csb 3900 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | |
| 11 | 8, 9, 10 | 3eqtr4i 2775 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 [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-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: cbvsum 15731 cbvprod 15949 measiuns 34218 poimirlem26 37653 climinf2mpt 45729 climinfmpt 45730 |
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