Theorem exmidpeirce 16668

Description: Excluded middle is equivalent to Peirce's law. 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 dcfrompeirce 1494. (Contributed by Jim Kingdon, 23-Apr-2026.)

Assertion

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

Distinct variable group:   x, y

Proof of Theorem exmidpeirce

Step	Hyp	Ref	Expression
1		exmidexmid 4288	. . . . 5 |- (EXMID -> DECID x = 1o )
2		peircedc 921	. . . . 5 |- (DECID x = 1o -> ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
3	1, 2	syl 14	. . . 4 |- (EXMID -> ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
4	3	ralrimivw 2605	. . 3 |- (EXMID -> A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
5	4	ralrimivw 2605	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
6		eqeq1 2237	. . . . . . . . 9 |- ( y = (/) -> ( y = 1o <-> (/) = 1o ) )
7	6	imbi2d 230	. . . . . . . 8 |- ( y = (/) -> ( ( x = 1o -> y = 1o ) <-> ( x = 1o -> (/) = 1o ) ) )
8	7	imbi1d 231	. . . . . . 7 |- ( y = (/) -> ( ( ( x = 1o -> y = 1o ) -> x = 1o ) <-> ( ( x = 1o -> (/) = 1o ) -> x = 1o ) ) )
9	8	imbi1d 231	. . . . . 6 |- ( y = (/) -> ( ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) <-> ( ( ( x = 1o -> (/) = 1o ) -> x = 1o ) -> x = 1o ) ) )
10		id 19	. . . . . 6 |- ( A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
11		0elpw 4256	. . . . . . 7 |- (/) e. ~P 1o
12	11	a1i 9	. . . . . 6 |- ( A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> (/) e. ~P 1o )
13	9, 10, 12	rspcdva 2914	. . . . 5 |- ( A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> ( ( ( x = 1o -> (/) = 1o ) -> x = 1o ) -> x = 1o ) )
14		1n0 6605	. . . . . . . . 9 |- 1o =/= (/)
15	14	nesymi 2447	. . . . . . . 8 |- -. (/) = 1o
16		mtt 691	. . . . . . . 8 |- ( -. (/) = 1o -> ( -. x = 1o <-> ( x = 1o -> (/) = 1o ) ) )
17	15, 16	ax-mp 5	. . . . . . 7 |- ( -. x = 1o <-> ( x = 1o -> (/) = 1o ) )
18	17	imbi1i 238	. . . . . 6 |- ( ( -. x = 1o -> x = 1o ) <-> ( ( x = 1o -> (/) = 1o ) -> x = 1o ) )
19	18	imbi1i 238	. . . . 5 |- ( ( ( -. x = 1o -> x = 1o ) -> x = 1o ) <-> ( ( ( x = 1o -> (/) = 1o ) -> x = 1o ) -> x = 1o ) )
20	13, 19	sylibr 134	. . . 4 |- ( A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> ( ( -. x = 1o -> x = 1o ) -> x = 1o ) )
21	20	ralimi 2594	. . 3 |- ( A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> A. x e. ~P 1o ( ( -. x = 1o -> x = 1o ) -> x = 1o ) )
22		jarl 664	. . . . 5 |- ( ( ( -. x = 1o -> x = 1o ) -> x = 1o ) -> ( -. -. x = 1o -> x = 1o ) )
23	22	ralimi 2594	. . . 4 |- ( A. x e. ~P 1o ( ( -. x = 1o -> x = 1o ) -> x = 1o ) -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
24		exmidnotnotr 16666	. . . 4 |- (EXMID <-> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
25	23, 24	sylibr 134	. . 3 |- ( A. x e. ~P 1o ( ( -. x = 1o -> x = 1o ) -> x = 1o ) -> EXMID )
26	21, 25	syl 14	. 2 |- ( A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> EXMID )
27	5, 26	impbii 126	1 |- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 841   = wceq 1397   e. wcel 2201   A.wral 2509   (/)c0 3493   ~Pcpw 3653  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: (None)