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Mirrors > Home > ILE Home > Th. List > difeq2i | GIF version |
Description: Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
Ref | Expression |
---|---|
difeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
difeq2i | ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | difeq2 3152 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 ∖ cdif 3032 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-ral 2393 df-rab 2397 df-dif 3037 |
This theorem is referenced by: difeq12i 3156 inssddif 3281 difdif2ss 3297 dif32 3303 difabs 3304 symdif1 3305 notrab 3317 dif0 3397 difdifdirss 3411 dfif3 3451 difpr 3626 dif1o 6287 unfiin 6765 |
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