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Mirrors > Home > ILE Home > Th. List > cbvab | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
cbvab.1 | ⊢ Ⅎ𝑦𝜑 |
cbvab.2 | ⊢ Ⅎ𝑥𝜓 |
cbvab.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvab | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvab.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | nfsb 1958 | . . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜓 |
3 | cbvab.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
4 | cbvab.3 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | equcoms 1719 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
6 | 5 | bicomd 141 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
7 | 3, 6 | sbie 1802 | . . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑) |
8 | sbequ 1851 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓)) | |
9 | 7, 8 | bitr3id 194 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓)) |
10 | 2, 9 | sbie 1802 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
11 | df-clab 2176 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
12 | df-clab 2176 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
13 | 10, 11, 12 | 3bitr4i 212 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
14 | 13 | eqriv 2186 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 Ⅎwnf 1471 [wsb 1773 ∈ wcel 2160 {cab 2175 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 |
This theorem is referenced by: cbvabv 2314 cbvrab 2750 cbvsbc 3006 cbvrabcsf 3137 dfdmf 4841 dfrnf 4889 funfvdm2f 5605 abrexex2g 6149 abrexex2 6153 |
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