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Theorem rabxm 3915
 Description: Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabxm 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabxm
StepHypRef Expression
1 rabid2 3096 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
2 exmidd 431 . . 3 (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑))
31, 2mprgbir 2911 . 2 𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4 unrab 3857 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
53, 4eqtr4i 2635 1 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977  {crab 2900   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-un 3545 This theorem is referenced by:  elnelun  3918  usgrafilem1  25940  esumrnmpt2  29457  ddemeas  29626  ballotth  29926  mbfposadd  32627  jm2.22  36580
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