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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 1939 | . . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
3 | df-clab 2127 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) | |
4 | df-clab 2127 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
5 | 2, 3, 4 | 3bitr4ri 212 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑}) |
6 | 5 | eqriv 2137 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑} |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 Ⅎwnf 1437 ∈ wcel 1481 [wsb 1736 {cab 2126 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 |
This theorem is referenced by: (None) |
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