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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 2276 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 |
3 | cbvcsb.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
4 | 3 | nfcri 2276 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 |
5 | cbvcsb.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
6 | 5 | eleq2d 2210 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
7 | 2, 4, 6 | cbvsbc 2941 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) |
8 | 7 | abbii 2256 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} |
9 | df-csb 3008 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
10 | df-csb 3008 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | |
11 | 8, 9, 10 | 3eqtr4i 2171 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 {cab 2126 Ⅎwnfc 2269 [wsbc 2913 ⦋csb 3007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-sbc 2914 df-csb 3008 |
This theorem is referenced by: cbvcsbv 3013 cbvsum 11161 |
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