Theorem exmidnotnotr 16952

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 1497. (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 4331	. . . 4 |- (EXMID -> DECID x = 1o )
2		notnotrdc 855	. . . 4 |- (DECID x = 1o -> ( -. -. x = 1o -> x = 1o ) )
3	1, 2	syl 14	. . 3 |- (EXMID -> ( -. -. x = 1o -> x = 1o ) )
4	3	ralrimivw 2624	. 2 |- (EXMID -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
5		eqeq1 2245	. . . . . . . . 9 |- ( x = y -> ( x = 1o <-> y = 1o ) )
6	5	notbid 677	. . . . . . . 8 |- ( x = y -> ( -. x = 1o <-> -. y = 1o ) )
7	6	notbid 677	. . . . . . 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 3695	. . . . . . . 8 |- ( y e. ~P 1o <-> y C_ 1o )
11		df1o2 6694	. . . . . . . . 9 |- 1o = { (/) }
12	11	sseq2i 3275	. . . . . . . 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 2934	. . . . 5 |- ( ( A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) /\ y C_ { (/) } ) -> ( -. -. y = 1o -> y = 1o ) )
16		df-stab 843	. . . . 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 2249	. . . . . 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 844	. . . 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 4343	. 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 842  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   (/)c0 3520   ~Pcpw 3688   {csn 3708  EXMIDwem 4329   1oc1o 6673

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 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 4247  ax-nul 4257  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  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 3714  df-exmid 4330  df-suc 4514  df-1o 6680

This theorem is referenced by:  exmidcon  16953  exmidpeirce  16954