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Theorem rabxmdc 3469
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 837 . . . . . 6 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
21a1d 22 . . . . 5 (DECID 𝜑 → (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
32alimi 1466 . . . 4 (∀𝑥DECID 𝜑 → ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
4 df-ral 2473 . . . 4 (∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
53, 4sylibr 134 . . 3 (∀𝑥DECID 𝜑 → ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
6 rabid2 2667 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
75, 6sylibr 134 . 2 (∀𝑥DECID 𝜑𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8 unrab 3421 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
97, 8eqtr4di 2240 1 (∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 709  DECID wdc 835  wal 1362   = wceq 1364  wcel 2160  wral 2468  {crab 2472  cun 3142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-un 3148
This theorem is referenced by: (None)
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