Theorem exmidexmid 4157

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 829, peircedc 900, or condc 839.

(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 3213	. . 3 |- { z e. { (/) } | ph } C_ { (/) }
2		df-exmid 4156	. . . 4 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
3		p0ex 4149	. . . . . 6 |- { (/) } e. _V
4	3	rabex 4108	. . . . 5 |- { z e. { (/) } | ph } e. _V
5		sseq1 3151	. . . . . 6 |- ( x = { z e. { (/) } | ph } -> ( x C_ { (/) } <-> { z e. { (/) } | ph } C_ { (/) } ) )
6		eleq2 2221	. . . . . . 7 |- ( x = { z e. { (/) } | ph } -> ( (/) e. x <-> (/) e. { z e. { (/) } | ph } ) )
7	6	dcbid 824	. . . . . 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 2806	. . . 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 4091	. . . . 5 |- (/) e. _V
13	12	snid 3591	. . . 4 |- (/) e. { (/) }
14		biidd 171	. . . . 5 |- ( z = (/) -> ( ph <-> ph ) )
15	14	elrab 2868	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ( (/) e. { (/) } /\ ph ) )
16	13, 15	mpbiran 925	. . 3 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
17	16	dcbii 826	. 2 |- (DECID (/) e. { z e. { (/) } | ph } <-> DECID ph )
18	11, 17	sylib 121	1 |- (EXMID -> DECID ph )

Syntax hints:   -> wi 4  DECID wdc 820   A.wal 1333   = wceq 1335   e. wcel 2128   {crab 2439   C_ wss 3102   (/)c0 3394   {csn 3560  EXMIDwem 4155

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-exmid 4156

This theorem is referenced by:  exmidn0m  4162  exmid0el  4165  exmidel  4166  exmidundif  4167  exmidundifim  4168  sbthlemi3  6903  sbthlemi5  6905  sbthlemi6  6906  exmidomniim  7084  exmidfodomrlemim  7136  exmidontriimlem1  7156