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Theorem rabxm 4337
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 3379 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
2 exmidd 889 . . 3 (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑))
31, 2mprgbir 3150 . 2 𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4 unrab 4271 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
53, 4eqtr4i 2844 1 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 841   = wceq 1528  wcel 2105  {crab 3139  cun 3931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-un 3938
This theorem is referenced by:  elnelun  4340  vtxdgoddnumeven  27262  esumrnmpt2  31226  ddemeas  31394  ballotth  31694  mbfposadd  34820  jm2.22  39470
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