Theorem wexmiddc 17042

Description: Weak excluded middle expressed using WEXMID implies decidability of a negated proposition. (Contributed by Jim Kingdon, 30-Jul-2026.)

Assertion

Ref	Expression
wexmiddc	⊢ (WEXMID → DECID ¬ 𝜑)

Proof of Theorem wexmiddc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wexmid 17041	. . . 4 ⊢ (WEXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ 𝑥 = 1o ∨ ¬ ¬ 𝑥 = 1o))
2		1oex 6695	. . . . . 6 ⊢ 1o ∈ V
3		ssrab2 3333	. . . . . 6 ⊢ {𝑦 ∈ 1o ∣ 𝜑} ⊆ 1o
4	2, 3	elpwi2 4294	. . . . 5 ⊢ {𝑦 ∈ 1o ∣ 𝜑} ∈ 𝒫 1o
5		eqeq1 2245	. . . . . . . 8 ⊢ (𝑥 = {𝑦 ∈ 1o ∣ 𝜑} → (𝑥 = 1o ↔ {𝑦 ∈ 1o ∣ 𝜑} = 1o))
6	5	notbid 677	. . . . . . 7 ⊢ (𝑥 = {𝑦 ∈ 1o ∣ 𝜑} → (¬ 𝑥 = 1o ↔ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o))
7	6	notbid 677	. . . . . . 7 ⊢ (𝑥 = {𝑦 ∈ 1o ∣ 𝜑} → (¬ ¬ 𝑥 = 1o ↔ ¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o))
8	6, 7	orbi12d 805	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ 1o ∣ 𝜑} → ((¬ 𝑥 = 1o ∨ ¬ ¬ 𝑥 = 1o) ↔ (¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o)))
9	8	rspcv 2925	. . . . 5 ⊢ ({𝑦 ∈ 1o ∣ 𝜑} ∈ 𝒫 1o → (∀𝑥 ∈ 𝒫 1o(¬ 𝑥 = 1o ∨ ¬ ¬ 𝑥 = 1o) → (¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o)))
10	4, 9	ax-mp 5	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 1o(¬ 𝑥 = 1o ∨ ¬ ¬ 𝑥 = 1o) → (¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o))
11	1, 10	sylbi 121	. . 3 ⊢ (WEXMID → (¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o))
12		rabid1o 17034	. . . . 5 ⊢ ({𝑦 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)
13	12	notbii 678	. . . 4 ⊢ (¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ↔ ¬ 𝜑)
14	13	notbii 678	. . . 4 ⊢ (¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ↔ ¬ ¬ 𝜑)
15	13, 14	orbi12i 776	. . 3 ⊢ ((¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o ∨ ¬ ¬ {𝑦 ∈ 1o ∣ 𝜑} = 1o) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
16	11, 15	sylib 122	. 2 ⊢ (WEXMID → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
17		df-dc 847	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
18	16, 17	sylibr 134	1 ⊢ (WEXMID → DECID ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821  𝒫 cpw 3688  1_oc1o 6680  WEXMIDwwem 17040

This proof depends on 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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  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 3715  df-pr 3716  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513  df-suc 4516  df-1o 6687  df-wexmid 17041

This theorem is used by:  wexmiddiffi  17044