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Theorem dfdif2 3005
Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfdif2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfdif2
StepHypRef Expression
1 df-dif 2999 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
2 df-rab 2368 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
31, 2eqtr4i 2111 1 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1289  wcel 1438  {cab 2074  {crab 2363  cdif 2994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-rab 2368  df-dif 2999
This theorem is referenced by:  dfdif3  3108  difeq1  3109  difeq2  3110  nfdif  3119  difidALT  3349
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