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Mirrors > Home > ILE Home > Th. List > csbtt | GIF version |
Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
csbtt | ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 2909 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | nfcr 2211 | . . . 4 ⊢ (Ⅎ𝑥𝐵 → Ⅎ𝑥 𝑦 ∈ 𝐵) | |
3 | sbctt 2880 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥 𝑦 ∈ 𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | sylan2 280 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) |
5 | 4 | abbi1dv 2198 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐵) |
6 | 1, 5 | syl5eq 2125 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 {cab 2067 Ⅎwnfc 2206 [wsbc 2815 ⦋csb 2908 |
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-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: csbconstgf 2919 sbnfc2 2962 |
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