Theorem exmidcon 16767

Description: Excluded middle is equivalent to the form of contraposition which removes negation. 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 dcfromcon 1494. (Contributed by Jim Kingdon, 22-Apr-2026.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidcon

Step	Hyp	Ref	Expression
1		exmidexmid 4308	. . . . 5 ⊢ (EXMID → DECID 𝑦 = 1o)
2		condc 861	. . . . 5 ⊢ (DECID 𝑦 = 1o → ((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
3	1, 2	syl 14	. . . 4 ⊢ (EXMID → ((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
4	3	ralrimivw 2616	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
5	4	ralrimivw 2616	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
6		eqeq1 2239	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 1o ↔ 1o = 1o))
7	6	notbid 673	. . . . . . 7 ⊢ (𝑥 = 1o → (¬ 𝑥 = 1o ↔ ¬ 1o = 1o))
8	7	imbi2d 230	. . . . . 6 ⊢ (𝑥 = 1o → ((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) ↔ (¬ 𝑦 = 1o → ¬ 1o = 1o)))
9	6	imbi1d 231	. . . . . 6 ⊢ (𝑥 = 1o → ((𝑥 = 1o → 𝑦 = 1o) ↔ (1o = 1o → 𝑦 = 1o)))
10	8, 9	imbi12d 234	. . . . 5 ⊢ (𝑥 = 1o → (((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)) ↔ ((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o))))
11	10	ralbidv 2542	. . . 4 ⊢ (𝑥 = 1o → (∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)) ↔ ∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o))))
12		id 19	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)) → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
13		1oex 6654	. . . . . 6 ⊢ 1o ∈ V
14	13	pwid 3686	. . . . 5 ⊢ 1o ∈ 𝒫 1o
15	14	a1i 9	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)) → 1o ∈ 𝒫 1o)
16	11, 12, 15	rspcdva 2925	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)) → ∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o)))
17		eqid 2232	. . . . . . 7 ⊢ 1o = 1o
18		ax-in2 620	. . . . . . . . 9 ⊢ (¬ ¬ 𝑦 = 1o → (¬ 𝑦 = 1o → ¬ 1o = 1o))
19	18	adantr 276	. . . . . . . 8 ⊢ ((¬ ¬ 𝑦 = 1o ∧ ((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o))) → (¬ 𝑦 = 1o → ¬ 1o = 1o))
20		simpr 110	. . . . . . . 8 ⊢ ((¬ ¬ 𝑦 = 1o ∧ ((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o))) → ((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o)))
21	19, 20	mpd 13	. . . . . . 7 ⊢ ((¬ ¬ 𝑦 = 1o ∧ ((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o))) → (1o = 1o → 𝑦 = 1o))
22	17, 21	mpi 15	. . . . . 6 ⊢ ((¬ ¬ 𝑦 = 1o ∧ ((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o))) → 𝑦 = 1o)
23	22	expcom 116	. . . . 5 ⊢ (((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o)) → (¬ ¬ 𝑦 = 1o → 𝑦 = 1o))
24	23	ralimi 2605	. . . 4 ⊢ (∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o)) → ∀𝑦 ∈ 𝒫 1o(¬ ¬ 𝑦 = 1o → 𝑦 = 1o))
25		exmidnotnotr 16766	. . . 4 ⊢ (EXMID ↔ ∀𝑦 ∈ 𝒫 1o(¬ ¬ 𝑦 = 1o → 𝑦 = 1o))
26	24, 25	sylibr 134	. . 3 ⊢ (∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 1o = 1o) → (1o = 1o → 𝑦 = 1o)) → EXMID)
27	16, 26	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)) → EXMID)
28	5, 27	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 842   = wceq 1398   ∈ wcel 2203  ∀wral 2520  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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553

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-rex 2526  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-pr 3695  df-uni 3914  df-tr 4208  df-exmid 4307  df-iord 4486  df-on 4488  df-suc 4491  df-1o 6646

This theorem is referenced by: (None)