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Theorem difeq2 3188
 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 2203 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21notbid 656 . . 3 (𝐴 = 𝐵 → (¬ 𝑥𝐴 ↔ ¬ 𝑥𝐵))
32rabbidv 2675 . 2 (𝐴 = 𝐵 → {𝑥𝐶 ∣ ¬ 𝑥𝐴} = {𝑥𝐶 ∣ ¬ 𝑥𝐵})
4 dfdif2 3079 . 2 (𝐶𝐴) = {𝑥𝐶 ∣ ¬ 𝑥𝐴}
5 dfdif2 3079 . 2 (𝐶𝐵) = {𝑥𝐶 ∣ ¬ 𝑥𝐵}
63, 4, 53eqtr4g 2197 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ∈ wcel 1480  {crab 2420   ∖ cdif 3068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rab 2425  df-dif 3073 This theorem is referenced by:  difeq12  3189  difeq2i  3191  difeq2d  3194  disjdif2  3441  ssdifeq0  3445  2oconcl  6339  diffitest  6784  diffifi  6791  undifdc  6815  sbthlem2  6849  isbth  6858  difinfinf  6989  ismkvnex  7032  iscld  12298
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