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Mirrors > Home > ILE Home > Th. List > nfrabxy | GIF version |
Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.) |
Ref | Expression |
---|---|
nfrabxy.1 | ⊢ Ⅎ𝑥𝜑 |
nfrabxy.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfrabxy | ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2357 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | nfrabxy.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2213 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfrabxy.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 3, 4 | nfan 1497 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
6 | 5 | nfab 2223 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
7 | 1, 6 | nfcxfr 2216 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 Ⅎwnf 1389 ∈ wcel 1433 {cab 2067 Ⅎwnfc 2206 {crab 2352 |
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-rab 2357 |
This theorem is referenced by: nfdif 3093 nfin 3172 nfse 4096 mpt2xopoveq 5878 nfsup 6405 caucvgprprlemaddq 6898 |
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