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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 2306 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 |
3 | cbvcsb.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
4 | 3 | nfcri 2306 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 |
5 | cbvcsb.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
6 | 5 | eleq2d 2240 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
7 | 2, 4, 6 | cbvsbc 2983 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) |
8 | 7 | abbii 2286 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} |
9 | df-csb 3050 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
10 | df-csb 3050 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | |
11 | 8, 9, 10 | 3eqtr4i 2201 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 {cab 2156 Ⅎwnfc 2299 [wsbc 2955 ⦋csb 3049 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-sbc 2956 df-csb 3050 |
This theorem is referenced by: cbvcsbv 3055 cbvsum 11312 |
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