Theorem exmidnotnotr 16766

Description: Excluded middle is equivalent to double negation elimination. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromnotnotr 1493. (Contributed by Jim Kingdon, 22-Apr-2026.)

Assertion

Ref	Expression
exmidnotnotr	|- (EXMID <-> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )

Proof of Theorem exmidnotnotr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidexmid 4308	. . . 4 |- (EXMID -> DECID x = 1o )
2		notnotrdc 851	. . . 4 |- (DECID x = 1o -> ( -. -. x = 1o -> x = 1o ) )
3	1, 2	syl 14	. . 3 |- (EXMID -> ( -. -. x = 1o -> x = 1o ) )
4	3	ralrimivw 2616	. 2 |- (EXMID -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
5		eqeq1 2239	. . . . . . . . 9 |- ( x = y -> ( x = 1o <-> y = 1o ) )
6	5	notbid 673	. . . . . . . 8 |- ( x = y -> ( -. x = 1o <-> -. y = 1o ) )
7	6	notbid 673	. . . . . . 7 |- ( x = y -> ( -. -. x = 1o <-> -. -. y = 1o ) )
8	7, 5	imbi12d 234	. . . . . 6 |- ( x = y -> ( ( -. -. x = 1o -> x = 1o ) <-> ( -. -. y = 1o -> y = 1o ) ) )
9		simpl 109	. . . . . 6 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
10		velpw 3675	. . . . . . . 8 |- ( y e. ~P 1o <-> y C_ 1o )
11		df1o2 6660	. . . . . . . . 9 |- 1o = { (/) }
12	11	sseq2i 3264	. . . . . . . 8 |- ( y C_ 1o <-> y C_ { (/) } )
13	10, 12	sylbbr 136	. . . . . . 7 |- ( y C_ { (/) } -> y e. ~P 1o )
14	13	adantl 277	. . . . . 6 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> y e. ~P 1o )
15	8, 9, 14	rspcdva 2925	. . . . 5 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> ( -. -. y = 1o -> y = 1o ) )
16		df-stab 839	. . . . 5 |- (STAB y = 1o <-> ( -. -. y = 1o -> y = 1o ) )
17	15, 16	sylibr 134	. . . 4 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> STAB y = 1o )
18	11	eqeq2i 2243	. . . . . 6 |- ( y = 1o <-> y = { (/) } )
19	18	a1i 9	. . . . 5 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> ( y = 1o <-> y = { (/) } ) )
20	19	stbid 840	. . . 4 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> (STAB y = 1o <-> STAB y = { (/) } ) )
21	17, 20	mpbid 147	. . 3 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> STAB y = { (/) } )
22	21	exmid1stab 4320	. 2 |- ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) -> EXMID )
23	4, 22	impbii 126	1 |- (EXMID <-> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  STAB wstab 838  DECID wdc 842   = wceq 1398   e. wcel 2203   A.wral 2520   C_ wss 3210   (/)c0 3507   ~Pcpw 3668   {csn 3688  EXMIDwem 4306   1oc1o 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:  exmidcon  16767  exmidpeirce  16768