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Mirrors > Home > ILE Home > Th. List > nfsbcdw | GIF version |
Description: Version of nfsbcd 2956 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfsbcdw.1 | ⊢ Ⅎ𝑦𝜑 |
nfsbcdw.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfsbcdw.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfsbcdw | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 2938 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
2 | nfsbcdw.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfsbcdw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfsbcdw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfabdw 2318 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 2, 5 | nfeld 2315 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfxfrd 1455 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1440 ∈ wcel 2128 {cab 2143 Ⅎwnfc 2286 [wsbc 2937 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-sbc 2938 |
This theorem is referenced by: nfcsbw 3067 |
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