Theorem exmidpeirce 17020

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 1499. (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 4331	. . . . 5 |- (EXMID -> DECID x = 1o )
2		peircedc 926	. . . . 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 2624	. . 3 |- (EXMID -> A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
5	4	ralrimivw 2624	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
6		eqeq1 2245	. . . . . . . . 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 4299	. . . . . . 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 2934	. . . . 5 |- ( A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) -> ( ( ( x = 1o -> (/) = 1o ) -> x = 1o ) -> x = 1o ) )
14		1n0 6699	. . . . . . . . 9 |- 1o =/= (/)
15	14	nesymi 2466	. . . . . . . 8 |- -. (/) = 1o
16		mtt 696	. . . . . . . 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 2613	. . 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 668	. . . . 5 |- ( ( ( -. x = 1o -> x = 1o ) -> x = 1o ) -> ( -. -. x = 1o -> x = 1o ) )
23	22	ralimi 2613	. . . 4 |- ( A. x e. ~P 1o ( ( -. x = 1o -> x = 1o ) -> x = 1o ) -> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
24		exmidnotnotr 17018	. . . 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 846   = wceq 1402   e. wcel 2209   A.wral 2528   (/)c0 3520   ~Pcpw 3688  EXMIDwem 4329   1oc1o 6674

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 6681

This theorem is referenced by: (None)