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Mirrors > Home > ILE Home > Th. List > difeq2d | GIF version |
Description: Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
Ref | Expression |
---|---|
difeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
difeq2d | ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | difeq2 3262 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∖ cdif 3141 |
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-in1 615 ax-in2 616 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 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-ral 2473 df-rab 2477 df-dif 3146 |
This theorem is referenced by: difeq12d 3269 exmid1stab 4223 phplem3 6872 phplem4 6873 phplem3g 6874 phplem4dom 6880 phplem4on 6885 fidifsnen 6888 xpfi 6947 sbthlem2 6975 sbthlemi3 6976 isbth 6984 ismkvnex 7171 setsvalg 12510 setsvala 12511 |
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