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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.) (Unnecessary distinct variable restrictions were removed 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 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
3 | nfopab.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | 2, 3 | nfan 1457 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 1528 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | 5 | nfex 1528 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | 6 | nfab 2182 | . 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
8 | 1, 7 | nfcxfr 2175 | 1 ⊢ Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 {cab 2026 Ⅎwnfc 2165 〈cop 3378 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-opab 3819 |
This theorem is referenced by: csbopabg 3835 nfmpt 3849 nfxp 4371 nfco 4501 nfcnv 4514 |
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