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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 4924 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) |
3 | reseqd.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | reseq2d 4925 | . 2 ⊢ (𝜑 → (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
5 | 2, 4 | eqtrd 2222 | 1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ↾ cres 4646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-opab 4080 df-xp 4650 df-res 4656 |
This theorem is referenced by: tfrlem3ag 6333 tfrlem3a 6334 tfrlemi1 6356 tfr1onlem3ag 6361 setsvalg 12541 znval 13929 psrval 13941 isxms 14403 isms 14405 |
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