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Theorem rabxm 4387
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 3462 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
2 exmidd 892 . . 3 (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑))
31, 2mprgbir 3066 . 2 𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4 unrab 4306 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
53, 4eqtr4i 2761 1 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 843   = wceq 1539  wcel 2104  {crab 3430  cun 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rab 3431  df-v 3474  df-un 3954
This theorem is referenced by:  elnelun  4390  vtxdgoddnumeven  29075  esumrnmpt2  33362  ddemeas  33530  ballotth  33832  mbfposadd  36840  jm2.22  42038
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