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Theorem rabxmdc 3445
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 831 . . . . . 6 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
21a1d 22 . . . . 5 (DECID 𝜑 → (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
32alimi 1448 . . . 4 (∀𝑥DECID 𝜑 → ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
4 df-ral 2453 . . . 4 (∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
53, 4sylibr 133 . . 3 (∀𝑥DECID 𝜑 → ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
6 rabid2 2646 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
75, 6sylibr 133 . 2 (∀𝑥DECID 𝜑𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8 unrab 3398 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
97, 8eqtr4di 2221 1 (∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 703  DECID wdc 829  wal 1346   = wceq 1348  wcel 2141  wral 2448  {crab 2452  cun 3119
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-un 3125
This theorem is referenced by: (None)
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