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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 2379 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | nfrabxy.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2229 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfrabxy.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 3, 4 | nfan 1509 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
6 | 5 | nfab 2240 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
7 | 1, 6 | nfcxfr 2232 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1401 ∈ wcel 1445 {cab 2081 Ⅎwnfc 2222 {crab 2374 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rab 2379 |
This theorem is referenced by: nfdif 3136 nfin 3221 nfse 4192 elfvmptrab1 5433 mpt2xopoveq 6043 nfsup 6767 caucvgprprlemaddq 7364 |
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