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Mirrors > Home > ILE Home > Th. List > nfres | GIF version |
Description: Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
nfres.1 | ⊢ Ⅎ𝑥𝐴 |
nfres.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfres | ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4559 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | nfres.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfres.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2282 | . . . 4 ⊢ Ⅎ𝑥V | |
5 | 3, 4 | nfxp 4574 | . . 3 ⊢ Ⅎ𝑥(𝐵 × V) |
6 | 2, 5 | nfin 3287 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V)) |
7 | 1, 6 | nfcxfr 2279 | 1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2269 Vcvv 2689 ∩ cin 3075 × cxp 4545 ↾ cres 4549 |
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 df-opab 3998 df-xp 4553 df-res 4559 |
This theorem is referenced by: nfima 4897 nffrec 6301 |
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