Theorem exmidexmid 4198

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 843, peircedc 914, or condc 853.

(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 3242	. . 3 |- { z e. { (/) } | ph } C_ { (/) }
2		df-exmid 4197	. . . 4 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
3		p0ex 4190	. . . . . 6 |- { (/) } e. _V
4	3	rabex 4149	. . . . 5 |- { z e. { (/) } | ph } e. _V
5		sseq1 3180	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
6		eleq2 2241	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
7	6	dcbid 838	. . . . . 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 2833	. . . 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 4132	. . . . 5 |- (/) e. _V
13	12	snid 3625	. . . 4 |- (/) e. { (/) }
14		biidd 172	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
15	14	elrab 2895	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ( (/) e. { (/) } /\ ph ) )
16	13, 15	mpbiran 940	. . 3 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
17	16	dcbii 840	. 2 |- (DECID (/) e. { z e. { (/) } | ph } <-> DECID ph )
18	11, 17	sylib 122	1 |- (EXMID -> DECID ph )

Syntax hints:   -> wi 4  DECID wdc 834   A.wal 1351   = wceq 1353   e. wcel 2148   {crab 2459   C_ wss 3131   (/)c0 3424   {csn 3594  EXMIDwem 4196

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-exmid 4197

This theorem is referenced by:  exmidn0m  4203  exmid0el  4206  exmidel  4207  exmidundif  4208  exmidundifim  4209  sbthlemi3  6961  sbthlemi5  6963  sbthlemi6  6964  exmidomniim  7142  exmidfodomrlemim  7203  exmidontriimlem1  7223  exmidapne  7262