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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 5771 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
2 | nfv 1508 | . . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
3 | nfoprab.1 | . . . . . . 7 ⊢ Ⅎ𝑤𝜑 | |
4 | 2, 3 | nfan 1544 | . . . . . 6 ⊢ Ⅎ𝑤(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
5 | 4 | nfex 1616 | . . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
6 | 5 | nfex 1616 | . . . 4 ⊢ Ⅎ𝑤∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
7 | 6 | nfex 1616 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
8 | 7 | nfab 2284 | . 2 ⊢ Ⅎ𝑤{𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
9 | 1, 8 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 {cab 2123 Ⅎwnfc 2266 〈cop 3525 {coprab 5768 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-oprab 5771 |
This theorem is referenced by: nfmpo 5833 |
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