Theorem exmidexmid 4247

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 845, peircedc 916, or condc 855.

(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 3282	. . 3 |- { z e. { (/) } | ph } C_ { (/) }
2		df-exmid 4246	. . . 4 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
3		p0ex 4239	. . . . . 6 |- { (/) } e. _V
4	3	rabex 4195	. . . . 5 |- { z e. { (/) } | ph } e. _V
5		sseq1 3220	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
6		eleq2 2270	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
7	6	dcbid 840	. . . . . 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 2871	. . . 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 4178	. . . . 5 |- (/) e. _V
13	12	snid 3668	. . . 4 |- (/) e. { (/) }
14		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
15	14	elrab 2933	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ( (/) e. { (/) } /\ ph ) )
16	13, 15	mpbiran 943	. . 3 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
17	16	dcbii 842	. 2 |- (DECID (/) e. { z e. { (/) } | ph } <-> DECID ph )
18	11, 17	sylib 122	1 |- (EXMID -> DECID ph )

Syntax hints:   -> wi 4  DECID wdc 836   A.wal 1371   = wceq 1373   e. wcel 2177   {crab 2489   C_ wss 3170   (/)c0 3464   {csn 3637  EXMIDwem 4245

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-nul 4177  ax-pow 4225

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-exmid 4246

This theorem is referenced by:  exmidn0m  4252  exmid0el  4255  exmidel  4256  exmidundif  4257  exmidundifim  4258  exmidpw2en  7023  sbthlemi3  7075  sbthlemi5  7077  sbthlemi6  7078  exmidomniim  7257  exmidfodomrlemim  7324  exmidontriimlem1  7348  exmidapne  7387