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Theorem difeq1 3126
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem difeq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rabeq 2625 . 2 (𝐴 = 𝐵 → {𝑥𝐴 ∣ ¬ 𝑥𝐶} = {𝑥𝐵 ∣ ¬ 𝑥𝐶})
2 dfdif2 3021 . 2 (𝐴𝐶) = {𝑥𝐴 ∣ ¬ 𝑥𝐶}
3 dfdif2 3021 . 2 (𝐵𝐶) = {𝑥𝐵 ∣ ¬ 𝑥𝐶}
41, 2, 33eqtr4g 2152 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1296  wcel 1445  {crab 2374  cdif 3010
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-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-dif 3015
This theorem is referenced by:  difeq12  3128  difeq1i  3129  difeq1d  3132  uneqdifeqim  3387  diffitest  6683
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