Theorem exmidexmid 4225

Description: EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 844, peircedc 915, or condc 854.

(Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmidexmid	|- (EXMID -> DECID ph )

Proof of Theorem exmidexmid

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3264	. . 3 |- { z e. { (/) } | ph } C_ { (/) }
2		df-exmid 4224	. . . 4 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
3		p0ex 4217	. . . . . 6 |- { (/) } e. _V
4	3	rabex 4173	. . . . 5 |- { z e. { (/) } | ph } e. _V
5		sseq1 3202	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
6		eleq2 2257	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
7	6	dcbid 839	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> (DECID (/) e. x <-> DECID (/) e. { z e. { (/) } | ph } ) )
8	5, 7	imbi12d 234	. . . . 5 |- ( x = { z e. { (/) } | ph } -> ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( { z e. { (/) } | ph } C_ { (/) } -> DECID (/) e. { z e. { (/) } | ph } ) ) )
9	4, 8	spcv 2854	. . . 4 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) -> ( { z e. { (/) } | ph } C_ { (/) } -> DECID (/) e. { z e. { (/) } | ph } ) )
10	2, 9	sylbi 121	. . 3 |- (EXMID -> ( { z e. { (/) } | ph } C_ { (/) } -> DECID (/) e. { z e. { (/) } | ph } ) )
11	1, 10	mpi 15	. 2 |- (EXMID -> DECID (/) e. { z e. { (/) } | ph } )
12		0ex 4156	. . . . 5 |- (/) e. _V
13	12	snid 3649	. . . 4 |- (/) e. { (/) }
14		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
15	14	elrab 2916	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ( (/) e. { (/) } /\ ph ) )
16	13, 15	mpbiran 942	. . 3 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
17	16	dcbii 841	. 2 |- (DECID (/) e. { z e. { (/) } | ph } <-> DECID ph )
18	11, 17	sylib 122	1 |- (EXMID -> DECID ph )

Syntax hints:   -> wi 4  DECID wdc 835   A.wal 1362   = wceq 1364   e. wcel 2164   {crab 2476   C_ wss 3153   (/)c0 3446   {csn 3618  EXMIDwem 4223

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-exmid 4224

This theorem is referenced by:  exmidn0m  4230  exmid0el  4233  exmidel  4234  exmidundif  4235  exmidundifim  4236  exmidpw2en  6968  sbthlemi3  7018  sbthlemi5  7020  sbthlemi6  7021  exmidomniim  7200  exmidfodomrlemim  7261  exmidontriimlem1  7281  exmidapne  7320