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Theorem difeq2 3135
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem difeq2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2163 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21notbid 633 . . 3 (𝐴 = 𝐵 → (¬ 𝑥𝐴 ↔ ¬ 𝑥𝐵))
32rabbidv 2630 . 2 (𝐴 = 𝐵 → {𝑥𝐶 ∣ ¬ 𝑥𝐴} = {𝑥𝐶 ∣ ¬ 𝑥𝐵})
4 dfdif2 3029 . 2 (𝐶𝐴) = {𝑥𝐶 ∣ ¬ 𝑥𝐴}
5 dfdif2 3029 . 2 (𝐶𝐵) = {𝑥𝐶 ∣ ¬ 𝑥𝐵}
63, 4, 53eqtr4g 2157 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1299  wcel 1448  {crab 2379  cdif 3018
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 584  ax-in2 585  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-ral 2380  df-rab 2384  df-dif 3023
This theorem is referenced by:  difeq12  3136  difeq2i  3138  difeq2d  3141  disjdif2  3388  ssdifeq0  3392  2oconcl  6266  diffitest  6710  diffifi  6717  undifdc  6741  sbthlem2  6774  isbth  6783  difinfinf  6901  iscld  12054
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