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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 3888 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | nfre1 2520 | . . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfab 2324 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
4 | 1, 3 | nfcxfr 2316 | 1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 {cab 2163 Ⅎwnfc 2306 ∃wrex 2456 ∪ ciun 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-iun 3888 |
This theorem is referenced by: ssiun2s 3930 triun 4113 eliunxp 4765 opeliunxp2 4766 opeliunxp2f 6236 ixpf 6717 ctiunctlemfo 12432 |
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