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Mirrors > Home > ILE Home > Th. List > nfuni | GIF version |
Description: Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
nfuni.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfuni | ⊢ Ⅎ𝑥∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 3785 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfuni.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfv 1515 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
4 | 2, 3 | nfrexxy 2503 | . . 3 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 |
5 | 4 | nfab 2311 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} |
6 | 1, 5 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑥∪ 𝐴 |
Colors of variables: wff set class |
Syntax hints: {cab 2150 Ⅎwnfc 2293 ∃wrex 2443 ∪ cuni 3783 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-uni 3784 |
This theorem is referenced by: nfiota1 5149 nfrecs 6266 nfsup 6948 |
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