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Mirrors > Home > ILE Home > Th. List > difeq1d | GIF version |
Description: Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
Ref | Expression |
---|---|
difeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
difeq1d | ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | difeq1 3157 | . 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 1316 ∖ cdif 3038 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 df-dif 3043 |
This theorem is referenced by: difeq12d 3165 diftpsn3 3631 phplem4 6717 phplem3g 6718 phplem4on 6729 en2other2 7020 isstruct2im 11896 isstruct2r 11897 setsfun0 11922 cldval 12195 difopn 12204 cnclima 12319 |
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