Theorem rabxmdc 3276

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 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 777	. . . . . 6 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2	1	a1d 22	. . . . 5 ⊢ (DECID 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
3	2	alimi 1384	. . . 4 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
4		df-ral 2353	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
5	3, 4	sylibr 132	. . 3 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
6		rabid2 2530	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
7	5, 6	sylibr 132	. 2 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8		unrab 3235	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
9	7, 8	syl6eqr 2131	1 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  DECID wdc 775  ∀wal 1282   = wceq 1284   ∈ wcel 1433  ∀wral 2348  {crab 2352   ∪ cun 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-un 2977

This theorem is referenced by: (None)