Theorem rabxmdc 3544

Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Jim Kingdon, 17-Jun-2026.)

Assertion

Ref	Expression
rabxmdc	⊢ (∀𝑥 ∈ 𝐴 DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabxmdc

Step	Hyp	Ref	Expression
1		exmiddc 844	. . . 4 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2	1	ralimi 2607	. . 3 ⊢ (∀𝑥 ∈ 𝐴 DECID 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
3		rabid2 2723	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
4	2, 3	sylibr 134	. 2 ⊢ (∀𝑥 ∈ 𝐴 DECID 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
5		unrab 3496	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
6	4, 5	eqtr4di 2285	1 ⊢ (∀𝑥 ∈ 𝐴 DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 716  DECID wdc 842   = wceq 1398  ∀wral 2522  {crab 2526   ∪ cun 3212

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rab 2531  df-v 2817  df-un 3218

This theorem is referenced by:  ballotfilemth  13225