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 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 837	. . . . . 6 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2	1	a1d 22	. . . . 5 ⊢ (DECID 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
3	2	alimi 1466	. . . 4 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
4		df-ral 2473	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
5	3, 4	sylibr 134	. . 3 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
6		rabid2 2667	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
7	5, 6	sylibr 134	. 2 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8		unrab 3421	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
9	7, 8	eqtr4di 2240	1 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))

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)