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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 1758 | . 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐴)) | |
2 | 1 | abbi2dv 2283 | 1 ⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 [wsb 1749 ∈ wcel 2135 {cab 2150 |
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-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 |
This theorem is referenced by: sbcel12g 3055 sbceqg 3056 |
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