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Mirrors > Home > ILE Home > Th. List > cbvcsb | GIF version |
Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
cbvcsb.1 | ⊢ Ⅎ𝑦𝐶 |
cbvcsb.2 | ⊢ Ⅎ𝑥𝐷 |
cbvcsb.3 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cbvcsb | ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvcsb.1 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
2 | 1 | nfcri 2313 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 |
3 | cbvcsb.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
4 | 3 | nfcri 2313 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 |
5 | cbvcsb.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
6 | 5 | eleq2d 2247 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
7 | 2, 4, 6 | cbvsbc 2991 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) |
8 | 7 | abbii 2293 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} |
9 | df-csb 3058 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
10 | df-csb 3058 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | |
11 | 8, 9, 10 | 3eqtr4i 2208 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 {cab 2163 Ⅎwnfc 2306 [wsbc 2962 ⦋csb 3057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-sbc 2963 df-csb 3058 |
This theorem is referenced by: cbvcsbv 3063 cbvsum 11367 |
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