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Theorem difeq2 3033
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 2083 . . . 4 (A = B → (x Ax B))
21notbid 579 . . 3 (A = B → (¬ x A ↔ ¬ x B))
32rabbidv 2527 . 2 (A = B → {x 𝐶 ∣ ¬ x A} = {x 𝐶 ∣ ¬ x B})
4 dfdif2 2903 . 2 (𝐶A) = {x 𝐶 ∣ ¬ x A}
5 dfdif2 2903 . 2 (𝐶B) = {x 𝐶 ∣ ¬ x B}
63, 4, 53eqtr4g 2079 1 (A = B → (𝐶A) = (𝐶B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1228   wcel 1374  {crab 2288  cdif 2891
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-in1 532  ax-in2 533  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-rab 2293  df-dif 2897
This theorem is referenced by:  difeq12  3034  difeq2i  3036  difeq2d  3039  ssdifeq0  3282  2oconcl  5937
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