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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 1897 | . . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜓 |
3 | cbvab.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
4 | cbvab.3 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | equcoms 1667 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
6 | 5 | bicomd 140 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
7 | 3, 6 | sbie 1747 | . . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑) |
8 | sbequ 1794 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓)) | |
9 | 7, 8 | syl5bbr 193 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓)) |
10 | 2, 9 | sbie 1747 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
11 | df-clab 2102 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
12 | df-clab 2102 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
13 | 10, 11, 12 | 3bitr4i 211 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
14 | 13 | eqriv 2112 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 Ⅎwnf 1419 ∈ wcel 1463 [wsb 1718 {cab 2101 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 |
This theorem is referenced by: cbvabv 2238 cbvrab 2655 cbvsbc 2905 cbvrabcsf 3031 dfdmf 4692 dfrnf 4740 funfvdm2f 5440 abrexex2g 5972 abrexex2 5976 |
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