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Theorem difeq2 3261
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 2252 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21notbid 668 . . 3 (𝐴 = 𝐵 → (¬ 𝑥𝐴 ↔ ¬ 𝑥𝐵))
32rabbidv 2740 . 2 (𝐴 = 𝐵 → {𝑥𝐶 ∣ ¬ 𝑥𝐴} = {𝑥𝐶 ∣ ¬ 𝑥𝐵})
4 dfdif2 3151 . 2 (𝐶𝐴) = {𝑥𝐶 ∣ ¬ 𝑥𝐴}
5 dfdif2 3151 . 2 (𝐶𝐵) = {𝑥𝐶 ∣ ¬ 𝑥𝐵}
63, 4, 53eqtr4g 2246 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1363  wcel 2159  {crab 2471  cdif 3140
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-ral 2472  df-rab 2476  df-dif 3145
This theorem is referenced by:  difeq12  3262  difeq2i  3264  difeq2d  3267  disjdif2  3515  ssdifeq0  3519  2oconcl  6457  diffitest  6904  diffifi  6911  undifdc  6940  sbthlem2  6974  isbth  6983  difinfinf  7117  ismkvnex  7170  iscld  13986
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