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Mirrors > Home > ILE Home > Th. List > nfiu1 | GIF version |
Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.) |
Ref | Expression |
---|---|
nfiu1 | ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 3810 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | nfre1 2474 | . . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfab 2284 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
4 | 1, 3 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 {cab 2123 Ⅎwnfc 2266 ∃wrex 2415 ∪ ciun 3808 |
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-rex 2420 df-iun 3810 |
This theorem is referenced by: ssiun2s 3852 triun 4034 eliunxp 4673 opeliunxp2 4674 opeliunxp2f 6128 ixpf 6607 ctiunctlemfo 11941 |
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