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Theorem dfsymdif4 4047
 Description: Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.)
Assertion
Ref Expression
dfsymdif4 (𝐴𝐵) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfsymdif4
StepHypRef Expression
1 elsymdif 4046 . 2 (𝑥 ∈ (𝐴𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
21abbi2i 2915 1 (𝐴𝐵) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1653   ∈ wcel 2157  {cab 2785   △ csymdif 4040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-symdif 4041 This theorem is referenced by:  mbfeqalem1  23749
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