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Mirrors > Home > ILE Home > Th. List > reseq12d | GIF version |
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
reseqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
reseqd.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
reseq12d | ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseqd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | reseq1d 4708 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
3 | reseqd.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | reseq2d 4709 | . 2 ⊢ (𝜑 → (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
5 | 2, 4 | eqtrd 2120 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ↾ cres 4438 |
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-opab 3898 df-xp 4442 df-res 4448 |
This theorem is referenced by: tfrlem3ag 6066 tfrlem3a 6067 tfrlemi1 6089 tfr1onlem3ag 6094 |
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