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Theorem rabxmdc 3249
 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 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabxmdc
StepHypRef Expression
1 exmiddc 744 . . . . . 6 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
21a1d 22 . . . . 5 (DECID 𝜑 → (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
32alimi 1344 . . . 4 (∀𝑥DECID 𝜑 → ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
4 df-ral 2311 . . . 4 (∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
53, 4sylibr 137 . . 3 (∀𝑥DECID 𝜑 → ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
6 rabid2 2486 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
75, 6sylibr 137 . 2 (∀𝑥DECID 𝜑𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8 unrab 3208 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
97, 8syl6eqr 2090 1 (∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310   ∪ cun 2915 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-un 2922 This theorem is referenced by: (None)
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