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Mirrors > Home > ILE Home > Th. List > nfoprab2 | GIF version |
Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.) |
Ref | Expression |
---|---|
nfoprab2 | ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oprab 5778 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
2 | nfe1 1472 | . . . 4 ⊢ Ⅎ𝑦∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) | |
3 | 2 | nfex 1616 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
4 | 3 | nfab 2286 | . 2 ⊢ Ⅎ𝑦{𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
5 | 1, 4 | nfcxfr 2278 | 1 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 {cab 2125 Ⅎwnfc 2268 〈cop 3530 {coprab 5775 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-oprab 5778 |
This theorem is referenced by: ssoprab2b 5828 nfmpo2 5839 ovi3 5907 tposoprab 6177 |
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