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Theorem difeq1 3055
 Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem difeq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rabeq 2551 . 2 (𝐴 = 𝐵 → {𝑥𝐴 ∣ ¬ 𝑥𝐶} = {𝑥𝐵 ∣ ¬ 𝑥𝐶})
2 dfdif2 2926 . 2 (𝐴𝐶) = {𝑥𝐴 ∣ ¬ 𝑥𝐶}
3 dfdif2 2926 . 2 (𝐵𝐶) = {𝑥𝐵 ∣ ¬ 𝑥𝐶}
41, 2, 33eqtr4g 2097 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ∈ wcel 1393  {crab 2310   ∖ cdif 2914 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-dif 2920 This theorem is referenced by:  difeq12  3057  difeq1i  3058  difeq1d  3061  uneqdifeqim  3308  diffitest  6344
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