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Mirrors > Home > MPE Home > Th. List > sbab | Structured version Visualization version 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 2235 | . 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐴)) | |
2 | 1 | eqabdv 2859 | 1 ⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 [wsb 2059 ∈ wcel 2098 {cab 2701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 |
This theorem is referenced by: sbcel12 4401 sbceqg 4402 |
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