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Mirrors > Home > ILE Home > Th. List > nfopab | GIF version |
Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
nfopab.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfopab | ⊢ Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opab 3985 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
3 | nfopab.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | 2, 3 | nfan 1544 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 1616 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | 5 | nfex 1616 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | 6 | nfab 2284 | . 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
8 | 1, 7 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 {cab 2123 Ⅎwnfc 2266 〈cop 3525 {copab 3983 |
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-opab 3985 |
This theorem is referenced by: csbopabg 4001 nfmpt 4015 nfxp 4561 nfco 4699 nfcnv 4713 nfofr 5981 |
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