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Mirrors > Home > ILE Home > Th. List > reseq12i | GIF version |
Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
reseqi.1 | ⊢ 𝐴 = 𝐵 |
reseqi.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
reseq12i | ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseqi.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | reseq1i 4823 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) |
3 | reseqi.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | reseq2i 4824 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
5 | 2, 4 | eqtri 2161 | 1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ↾ 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-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-opab 3998 df-xp 4553 df-res 4559 |
This theorem is referenced by: cnvresid 5205 |
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