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Theorem rabxmdc 3394
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 821 . . . . . 6 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
21a1d 22 . . . . 5 (DECID 𝜑 → (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
32alimi 1431 . . . 4 (∀𝑥DECID 𝜑 → ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
4 df-ral 2421 . . . 4 (∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
53, 4sylibr 133 . . 3 (∀𝑥DECID 𝜑 → ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
6 rabid2 2607 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
75, 6sylibr 133 . 2 (∀𝑥DECID 𝜑𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8 unrab 3347 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
97, 8eqtr4di 2190 1 (∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 697  DECID wdc 819  wal 1329   = wceq 1331  wcel 1480  wral 2416  {crab 2420  cun 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-un 3075
This theorem is referenced by: (None)
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