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Theorem difeq2 3050
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 2098 . . . 4 (A = B → (x Ax B))
21notbid 591 . . 3 (A = B → (¬ x A ↔ ¬ x B))
32rabbidv 2543 . 2 (A = B → {x 𝐶 ∣ ¬ x A} = {x 𝐶 ∣ ¬ x B})
4 dfdif2 2920 . 2 (𝐶A) = {x 𝐶 ∣ ¬ x A}
5 dfdif2 2920 . 2 (𝐶B) = {x 𝐶 ∣ ¬ x B}
63, 4, 53eqtr4g 2094 1 (A = B → (𝐶A) = (𝐶B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  {crab 2304  cdif 2908
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 544  ax-in2 545  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rab 2309  df-dif 2914
This theorem is referenced by:  difeq12  3051  difeq2i  3053  difeq2d  3056  ssdifeq0  3299  2oconcl  5961
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