Theorem exmidexmid 4120

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 828, peircedc 899, or condc 838.

(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 3182	. . 3 |- { z e. { (/) } | ph } C_ { (/) }
2		df-exmid 4119	. . . 4 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
3		p0ex 4112	. . . . . 6 |- { (/) } e. _V
4	3	rabex 4072	. . . . 5 |- { z e. { (/) } | ph } e. _V
5		sseq1 3120	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
6		eleq2 2203	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
7	6	dcbid 823	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> (DECID (/) e. x <-> DECID (/) e. { z e. { (/) } | ph } ) )
8	5, 7	imbi12d 233	. . . . 5 |- ( x = { z e. { (/) } | ph } -> ( ( x C_ { (/) } -> DECID (/) e. x ) <-> ( { z e. { (/) } | ph } C_ { (/) } -> DECID (/) e. { z e. { (/) } | ph } ) ) )
9	4, 8	spcv 2779	. . . 4 |- ( A. x ( x C_ { (/) } -> DECID (/) e. x ) -> ( { z e. { (/) } | ph } C_ { (/) } -> DECID (/) e. { z e. { (/) } | ph } ) )
10	2, 9	sylbi 120	. . 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 4055	. . . . 5 |- (/) e. _V
13	12	snid 3556	. . . 4 |- (/) e. { (/) }
14		biidd 171	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
15	14	elrab 2840	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ( (/) e. { (/) } /\ ph ) )
16	13, 15	mpbiran 924	. . 3 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
17	16	dcbii 825	. 2 |- (DECID (/) e. { z e. { (/) } | ph } <-> DECID ph )
18	11, 17	sylib 121	1 |- (EXMID -> DECID ph )

Syntax hints:   -> wi 4  DECID wdc 819   A.wal 1329   = wceq 1331   e. wcel 1480   {crab 2420   C_ wss 3071   (/)c0 3363   {csn 3527  EXMIDwem 4118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119

This theorem is referenced by:  exmidn0m  4124  exmid0el  4127  exmidel  4128  exmidundif  4129  exmidundifim  4130  sbthlemi3  6847  sbthlemi5  6849  sbthlemi6  6850  exmidomniim  7013  exmidfodomrlemim  7057