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Mirrors > Home > ILE Home > Th. List > difeq1 | GIF version |
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
difeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq 2625 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶} = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶}) | |
2 | dfdif2 3021 | . 2 ⊢ (𝐴 ∖ 𝐶) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐶} | |
3 | dfdif2 3021 | . 2 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∈ 𝐵 ∣ ¬ 𝑥 ∈ 𝐶} | |
4 | 1, 2, 3 | 3eqtr4g 2152 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1296 ∈ wcel 1445 {crab 2374 ∖ cdif 3010 |
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-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rab 2379 df-dif 3015 |
This theorem is referenced by: difeq12 3128 difeq1i 3129 difeq1d 3132 uneqdifeqim 3387 diffitest 6683 |
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