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Mirrors > Home > ILE Home > Th. List > nfcsbd | GIF version |
Description: Deduction version of nfcsb 2940. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfcsbd.1 | ⊢ Ⅎ𝑦𝜑 |
nfcsbd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfcsbd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfcsbd | ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 2909 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵} | |
2 | nfv 1461 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfcsbd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfcsbd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfcsbd.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2232 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfsbcd 2834 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 2237 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2217 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 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-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: nfcsb 2940 |
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