Theorem exmidnotnotr 16666

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 1492. (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 4288	. . . 4 |- (EXMID -> DECID x = 1o )
2		notnotrdc 850	. . . 4 |- (DECID x = 1o -> ( -. -. x = 1o -> x = 1o ) )
3	1, 2	syl 14	. . 3 |- (EXMID -> ( -. -. x = 1o -> x = 1o ) )
4	3	ralrimivw 2605	. 2 |- (EXMID -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
5		eqeq1 2237	. . . . . . . . 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 3660	. . . . . . . 8 |- ( y e. ~P 1o <-> y C_ 1o )
11		df1o2 6601	. . . . . . . . 9 |- 1o = { (/) }
12	11	sseq2i 3253	. . . . . . . 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 2914	. . . . 5 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> ( -. -. y = 1o -> y = 1o ) )
16		df-stab 838	. . . . 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 2241	. . . . . 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 839	. . . 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 4300	. 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 837  DECID wdc 841   = wceq 1397   e. wcel 2201   A.wral 2509   C_ wss 3199   (/)c0 3493   ~Pcpw 3653   {csn 3670  EXMIDwem 4286   1oc1o 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:  exmidcon  16667  exmidpeirce  16668