Theorem exmidpeirce 16830

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 1495. (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 4311	. . . . 5 |- (EXMID -> DECID x = 1o )
2		peircedc 922	. . . . 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 2618	. . 3 |- (EXMID -> A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
5	4	ralrimivw 2618	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
6		eqeq1 2241	. . . . . . . . 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 4279	. . . . . . 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 2928	. . . . 5 |- ( A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> ( ( ( x = 1o -> (/) = 1o ) -> x = 1o ) -> x = 1o ) )
14		1n0 6667	. . . . . . . . 9 |- 1o =/= (/)
15	14	nesymi 2460	. . . . . . . 8 |- -. (/) = 1o
16		mtt 692	. . . . . . . 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 2607	. . 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 2607	. . . 4 |- ( A. x e. ~P 1o ( ( -. x = 1o -> x = 1o ) -> x = 1o ) -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
24		exmidnotnotr 16828	. . . 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 842   = wceq 1398   e. wcel 2205   A.wral 2522   (/)c0 3510   ~Pcpw 3671  EXMIDwem 4309   1oc1o 6642

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 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289

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 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-exmid 4310  df-suc 4494  df-1o 6649

This theorem is referenced by: (None)