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Mirrors > Home > ILE Home > Th. List > nfoprab | GIF version |
Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
nfoprab.1 | ⊢ Ⅎ𝑤𝜑 |
Ref | Expression |
---|---|
nfoprab | ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oprab 5878 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
2 | nfv 1528 | . . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
3 | nfoprab.1 | . . . . . . 7 ⊢ Ⅎ𝑤𝜑 | |
4 | 2, 3 | nfan 1565 | . . . . . 6 ⊢ Ⅎ𝑤(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
5 | 4 | nfex 1637 | . . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
6 | 5 | nfex 1637 | . . . 4 ⊢ Ⅎ𝑤∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
7 | 6 | nfex 1637 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
8 | 7 | nfab 2324 | . 2 ⊢ Ⅎ𝑤{𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
9 | 1, 8 | nfcxfr 2316 | 1 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 Ⅎwnf 1460 ∃wex 1492 {cab 2163 Ⅎwnfc 2306 〈cop 3595 {coprab 5875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-oprab 5878 |
This theorem is referenced by: nfmpo 5943 |
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