Theorem bj-d0clsepcl 10049

Description: Δ₀-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-d0clsepcl	|- DECID ph

Proof of Theorem bj-d0clsepcl

Dummy variables x a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 3884	. . . . . . 7 |- (/) e. _V
2	1	bj-snex 10033	. . . . . 6 |- { (/) } e. _V
3	2	zfauscl 3877	. . . . 5 |- E. a A. x ( x e. a <-> ( x e. { (/) } /\ ph ) )
4		eleq1 2100	. . . . . . 7 |- ( x = (/) -> ( x e. a <-> (/) e. a ) )
5		eleq1 2100	. . . . . . . 8 |- ( x = (/) -> ( x e. { (/) } <-> (/) e. { (/) } ) )
6	5	anbi1d 438	. . . . . . 7 |- ( x = (/) -> ( ( x e. { (/) } /\ ph ) <-> ( (/) e. { (/) } /\ ph ) ) )
7	4, 6	bibi12d 224	. . . . . 6 |- ( x = (/) -> ( ( x e. a <-> ( x e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) ) )
8	1, 7	spcv 2646	. . . . 5 |- ( A. x ( x e. a <-> ( x e. { (/) } /\ ph ) ) -> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) )
9	3, 8	eximii 1493	. . . 4 |- E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) )
10	1	snid 3402	. . . . . . . 8 |- (/) e. { (/) }
11	10	biantrur 287	. . . . . . 7 |- ( ph <-> ( (/) e. { (/) } /\ ph ) )
12	11	bicomi 123	. . . . . 6 |- ( ( (/) e. { (/) } /\ ph ) <-> ph )
13	12	bibi2i 216	. . . . 5 |- ( ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ph ) )
14	13	exbii 1496	. . . 4 |- ( E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> E. a ( (/) e. a <-> ph ) )
15	9, 14	mpbi 133	. . 3 |- E. a ( (/) e. a <-> ph )
16		bj-bd0el 9988	. . . . 5 |- BOUNDED (/) e. a
17	16	ax-bj-d0cl 10044	. . . 4 |- DECID (/) e. a
18		bj-dcbi 10048	. . . 4 |- ( ( (/) e. a <-> ph ) -> (DECID (/) e. a <-> DECID ph ) )
19	17, 18	mpbii 136	. . 3 |- ( ( (/) e. a <-> ph ) -> DECID ph )
20	15, 19	eximii 1493	. 2 |- E. aDECID ph
21		bj-ex 9902	. 2 |- ( E. aDECID ph -> DECID ph )
22	20, 21	ax-mp 7	1 |- DECID ph

Syntax hints:   /\ wa 97   <-> wb 98  DECID wdc 742   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   (/)c0 3224   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pr 3944  ax-bd0 9933  ax-bdim 9934  ax-bdor 9936  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdsep 10004  ax-bj-d0cl 10044

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-bdc 9961

This theorem is referenced by: (None)