Theorem bj-d0clsepcl 13807

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 4109	. . . . . . 7 |- (/) e. _V
2	1	bj-snex 13795	. . . . . 6 |- { (/) } e. _V
3	2	zfauscl 4102	. . . . 5 |- E. a A. x ( x e. a <-> ( x e. { (/) } /\ ph ) )
4		eleq1 2229	. . . . . . 7 |- ( x = (/) -> ( x e. a <-> (/) e. a ) )
5		eleq1 2229	. . . . . . . 8 |- ( x = (/) -> ( x e. { (/) } <-> (/) e. { (/) } ) )
6	5	anbi1d 461	. . . . . . 7 |- ( x = (/) -> ( ( x e. { (/) } /\ ph ) <-> ( (/) e. { (/) } /\ ph ) ) )
7	4, 6	bibi12d 234	. . . . . 6 |- ( x = (/) -> ( ( x e. a <-> ( x e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) ) )
8	1, 7	spcv 2820	. . . . 5 |- ( A. x ( x e. a <-> ( x e. { (/) } /\ ph ) ) -> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) )
9	3, 8	eximii 1590	. . . 4 |- E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) )
10	1	snid 3607	. . . . . . . 8 |- (/) e. { (/) }
11	10	biantrur 301	. . . . . . 7 |- ( ph <-> ( (/) e. { (/) } /\ ph ) )
12	11	bicomi 131	. . . . . 6 |- ( ( (/) e. { (/) } /\ ph ) <-> ph )
13	12	bibi2i 226	. . . . 5 |- ( ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ph ) )
14	13	exbii 1593	. . . 4 |- ( E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> E. a ( (/) e. a <-> ph ) )
15	9, 14	mpbi 144	. . 3 |- E. a ( (/) e. a <-> ph )
16		bj-bd0el 13750	. . . . 5 |- BOUNDED (/) e. a
17	16	ax-bj-d0cl 13806	. . . 4 |- DECID (/) e. a
18		dcbiit 829	. . . 4 |- ( ( (/) e. a <-> ph ) -> (DECID (/) e. a <-> DECID ph ) )
19	17, 18	mpbii 147	. . 3 |- ( ( (/) e. a <-> ph ) -> DECID ph )
20	15, 19	eximii 1590	. 2 |- E. aDECID ph
21		bj-ex 13643	. 2 |- ( E. aDECID ph -> DECID ph )
22	20, 21	ax-mp 5	1 |- DECID ph

Syntax hints:   /\ wa 103   <-> wb 104  DECID wdc 824   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   (/)c0 3409   {csn 3576

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pr 4187  ax-bd0 13695  ax-bdim 13696  ax-bdor 13698  ax-bdn 13699  ax-bdal 13700  ax-bdex 13701  ax-bdeq 13702  ax-bdsep 13766  ax-bj-d0cl 13806

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-bdc 13723

This theorem is referenced by: (None)