Theorem exmidpeirce 16768

Description: Excluded middle is equivalent to Peirce's law. Read an element of 𝒫 1_o as being a truth value and 𝑥 = 1_o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1_o, see dcfrompeirce 1495. (Contributed by Jim Kingdon, 23-Apr-2026.)

Assertion

Ref	Expression
exmidpeirce	⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidpeirce

Step	Hyp	Ref	Expression
1		exmidexmid 4308	. . . . 5 ⊢ (EXMID → DECID 𝑥 = 1o)
2		peircedc 922	. . . . 5 ⊢ (DECID 𝑥 = 1o → (((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
3	1, 2	syl 14	. . . 4 ⊢ (EXMID → (((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
4	3	ralrimivw 2616	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
5	4	ralrimivw 2616	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
6		eqeq1 2239	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 = 1o ↔ ∅ = 1o))
7	6	imbi2d 230	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑥 = 1o → 𝑦 = 1o) ↔ (𝑥 = 1o → ∅ = 1o)))
8	7	imbi1d 231	. . . . . . 7 ⊢ (𝑦 = ∅ → (((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) ↔ ((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o)))
9	8	imbi1d 231	. . . . . 6 ⊢ (𝑦 = ∅ → ((((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) ↔ (((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o) → 𝑥 = 1o)))
10		id 19	. . . . . 6 ⊢ (∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → ∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
11		0elpw 4276	. . . . . . 7 ⊢ ∅ ∈ 𝒫 1o
12	11	a1i 9	. . . . . 6 ⊢ (∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → ∅ ∈ 𝒫 1o)
13	9, 10, 12	rspcdva 2925	. . . . 5 ⊢ (∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → (((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
14		1n0 6664	. . . . . . . . 9 ⊢ 1o ≠ ∅
15	14	nesymi 2458	. . . . . . . 8 ⊢ ¬ ∅ = 1o
16		mtt 692	. . . . . . . 8 ⊢ (¬ ∅ = 1o → (¬ 𝑥 = 1o ↔ (𝑥 = 1o → ∅ = 1o)))
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ (¬ 𝑥 = 1o ↔ (𝑥 = 1o → ∅ = 1o))
18	17	imbi1i 238	. . . . . 6 ⊢ ((¬ 𝑥 = 1o → 𝑥 = 1o) ↔ ((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o))
19	18	imbi1i 238	. . . . 5 ⊢ (((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) ↔ (((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
20	13, 19	sylibr 134	. . . 4 ⊢ (∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → ((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o))
21	20	ralimi 2605	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → ∀𝑥 ∈ 𝒫 1o((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o))
22		jarl 664	. . . . 5 ⊢ (((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) → (¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
23	22	ralimi 2605	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 1o((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) → ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
24		exmidnotnotr 16766	. . . 4 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
25	23, 24	sylibr 134	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) → EXMID)
26	21, 25	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → EXMID)
27	5, 26	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 842   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∅c0 3507  𝒫 cpw 3668  EXMIDwem 4306  1_oc1o 6639

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-in1 619  ax-in2 620  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-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-exmid 4307  df-suc 4491  df-1o 6646

This theorem is referenced by: (None)