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Theorem rabxmdc 3277
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 755 . . . . . 6 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
21a1d 22 . . . . 5 (DECID 𝜑 → (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
32alimi 1360 . . . 4 (∀𝑥DECID 𝜑 → ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
4 df-ral 2328 . . . 4 (∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝜑 ∨ ¬ 𝜑)))
53, 4sylibr 141 . . 3 (∀𝑥DECID 𝜑 → ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
6 rabid2 2503 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
75, 6sylibr 141 . 2 (∀𝑥DECID 𝜑𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8 unrab 3236 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
97, 8syl6eqr 2106 1 (∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 639  DECID wdc 753  wal 1257   = wceq 1259  wcel 1409  wral 2323  {crab 2327  cun 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-un 2950
This theorem is referenced by: (None)
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