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Theorem rabxmdc 3243
Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
Assertion
Ref Expression
rabxmdc (xDECID φA = ({x Aφ} ∪ {x A ∣ ¬ φ}))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rabxmdc
StepHypRef Expression
1 exmiddc 743 . . . . . 6 (DECID φ → (φ ¬ φ))
21a1d 22 . . . . 5 (DECID φ → (x A → (φ ¬ φ)))
32alimi 1341 . . . 4 (xDECID φx(x A → (φ ¬ φ)))
4 df-ral 2305 . . . 4 (x A (φ ¬ φ) ↔ x(x A → (φ ¬ φ)))
53, 4sylibr 137 . . 3 (xDECID φx A (φ ¬ φ))
6 rabid2 2480 . . 3 (A = {x A ∣ (φ ¬ φ)} ↔ x A (φ ¬ φ))
75, 6sylibr 137 . 2 (xDECID φA = {x A ∣ (φ ¬ φ)})
8 unrab 3202 . 2 ({x Aφ} ∪ {x A ∣ ¬ φ}) = {x A ∣ (φ ¬ φ)}
97, 8syl6eqr 2087 1 (xDECID φA = ({x Aφ} ∪ {x A ∣ ¬ φ}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 628  DECID wdc 741  wal 1240   = wceq 1242   wcel 1390  wral 2300  {crab 2304  cun 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-un 2916
This theorem is referenced by: (None)
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