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Mirrors > Home > ILE Home > Th. List > nfin | GIF version |
Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfin.1 | ⊢ Ⅎ𝑥𝐴 |
nfin.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfin | ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3083 | . 2 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵} | |
2 | nfin.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 2 | nfcri 2276 | . . 3 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
4 | nfin.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | nfrabxy 2614 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵} |
6 | 1, 5 | nfcxfr 2279 | 1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 Ⅎwnfc 2269 {crab 2421 ∩ cin 3075 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-in 3082 |
This theorem is referenced by: csbing 3288 nfres 4829 |
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