Theorem exmidpeirce 17020

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 1499. (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 4331	. . . . 5 ⊢ (EXMID → DECID 𝑥 = 1o)
2		peircedc 926	. . . . 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 2624	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
5	4	ralrimivw 2624	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
6		eqeq1 2245	. . . . . . . . 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 4299	. . . . . . 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 2934	. . . . 5 ⊢ (∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → (((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
14		1n0 6699	. . . . . . . . 9 ⊢ 1o ≠ ∅
15	14	nesymi 2466	. . . . . . . 8 ⊢ ¬ ∅ = 1o
16		mtt 696	. . . . . . . 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 2613	. . 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 668	. . . . 5 ⊢ (((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) → (¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
23	22	ralimi 2613	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 1o((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) → ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
24		exmidnotnotr 17018	. . . 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 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∅c0 3520  𝒫 cpw 3688  EXMIDwem 4329  1_oc1o 6674

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-exmid 4330  df-suc 4514  df-1o 6681

This theorem is referenced by: (None)