Theorem exmidpeirce 16668

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 1494. (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 4288	. . . . 5 ⊢ (EXMID → DECID 𝑥 = 1o)
2		peircedc 921	. . . . 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 2605	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
5	4	ralrimivw 2605	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
6		eqeq1 2237	. . . . . . . . 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 4256	. . . . . . 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 2914	. . . . 5 ⊢ (∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o) → (((𝑥 = 1o → ∅ = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
14		1n0 6605	. . . . . . . . 9 ⊢ 1o ≠ ∅
15	14	nesymi 2447	. . . . . . . 8 ⊢ ¬ ∅ = 1o
16		mtt 691	. . . . . . . 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 2594	. . 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 2594	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 1o((¬ 𝑥 = 1o → 𝑥 = 1o) → 𝑥 = 1o) → ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
24		exmidnotnotr 16666	. . . 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 841   = wceq 1397   ∈ wcel 2201  ∀wral 2509  ∅c0 3493  𝒫 cpw 3653  EXMIDwem 4286  1_oc1o 6580

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-exmid 4287  df-suc 4470  df-1o 6587

This theorem is referenced by: (None)