Theorem exmidexmid 4309

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 851, peircedc 922, or condc 861.

(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 3323	. . 3 |- { z e. { (/) } | ph } C_ { (/) }
2		df-exmid 4308	. . . 4 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
3		p0ex 4301	. . . . . 6 |- { (/) } e. _V
4	3	rabex 4256	. . . . 5 |- { z e. { (/) } | ph } e. _V
5		sseq1 3261	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
6		eleq2 2296	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
7	6	dcbid 846	. . . . . 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 2911	. . . 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 4237	. . . . 5 |- (/) e. _V
13	12	snid 3720	. . . 4 |- (/) e. { (/) }
14		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
15	14	elrab 2973	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ( (/) e. { (/) } /\ ph ) )
16	13, 15	mpbiran 949	. . 3 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
17	16	dcbii 848	. 2 |- (DECID (/) e. { z e. { (/) } | ph } <-> DECID ph )
18	11, 17	sylib 122	1 |- (EXMID -> DECID ph )

Syntax hints:   -> wi 4  DECID wdc 842   A.wal 1396   = wceq 1398   e. wcel 2203   {crab 2524   C_ wss 3211   (/)c0 3508   {csn 3689  EXMIDwem 4307

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 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-exmid 4308

This theorem is referenced by:  exmidn0m  4314  exmid0el  4317  exmidel  4318  exmidundif  4319  exmidundifim  4320  exmidpw2en  7172  exmidssfi  7199  sbthlemi3  7229  sbthlemi5  7231  sbthlemi6  7232  exmidomniim  7432  exmidfodomrlemim  7504  exmidontriimlem1  7528  exmidapne  7574  pw1dceq  16778  exmidnotnotr  16779  exmidcon  16780  exmidpeirce  16781