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Mirrors > Home > ILE Home > Th. List > reseq1d | GIF version |
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
reseqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
reseq1d | ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | reseq1 4707 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ↾ cres 4440 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-in 3005 df-res 4450 |
This theorem is referenced by: reseq12d 4714 fun2ssres 5057 funcnvres2 5089 funimaexg 5098 fresin 5189 offres 5906 tfrlemisucaccv 6090 tfrlemi1 6097 tfr1onlemsucaccv 6106 tfrcllemsucaccv 6119 freceq1 6157 freceq2 6158 fseq1p1m1 9504 setsresg 11527 setscom 11529 |
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