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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 1822 | . . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜓 |
3 | cbvab.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
4 | cbvab.3 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | equcoms 1594 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
6 | 5 | bicomd 129 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
7 | 3, 6 | sbie 1674 | . . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑) |
8 | sbequ 1721 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓)) | |
9 | 7, 8 | syl5bbr 183 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓)) |
10 | 2, 9 | sbie 1674 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
11 | df-clab 2027 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
12 | df-clab 2027 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
13 | 10, 11, 12 | 3bitr4i 201 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
14 | 13 | eqriv 2037 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 [wsb 1645 {cab 2026 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 |
This theorem is referenced by: cbvabv 2161 cbvrab 2555 cbvsbc 2791 cbvrabcsf 2911 dfdmf 4528 dfrnf 4575 funfvdm2f 5238 abrexex2g 5747 abrexex2 5751 |
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