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Mirrors > Home > ILE Home > Th. List > sbab | GIF version |
Description: The right-hand side of the second equality is a way of representing proper substitution of 𝑦 for 𝑥 into a class variable. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
sbab | ⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 1725 | . 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐴)) | |
2 | 1 | abbi2dv 2231 | 1 ⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 [wsb 1716 {cab 2099 |
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-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 |
This theorem is referenced by: sbcel12g 2982 sbceqg 2983 |
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