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Mirrors > Home > ILE Home > Th. List > sb8ab | GIF version |
Description: Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.) |
Ref | Expression |
---|---|
sb8ab.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8ab | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8ab.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sbco2 1880 | . . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
3 | df-clab 2068 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) | |
4 | df-clab 2068 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
5 | 2, 3, 4 | 3bitr4ri 211 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑}) |
6 | 5 | eqriv 2078 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑} |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 [wsb 1685 {cab 2067 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 |
This theorem is referenced by: (None) |
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