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Theorem dfdif2 3085
 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 3079 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
2 df-rab 2426 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
31, 2eqtr4i 2164 1 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {cab 2126  {crab 2421   ∖ cdif 3074 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-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-rab 2426  df-dif 3079 This theorem is referenced by:  dfdif3  3192  difeq1  3193  difeq2  3194  nfdif  3203  difidALT  3438
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