Theorem bj-d0clsepcl 10863

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 3907	. . . . . . 7 |- (/) e. _V
2	1	bj-snex 10847	. . . . . 6 |- { (/) } e. _V
3	2	zfauscl 3900	. . . . 5 |- E. a A. x ( x e. a <-> ( x e. { (/) } /\ ph ) )
4		eleq1 2142	. . . . . . 7 |- ( x = (/) -> ( x e. a <-> (/) e. a ) )
5		eleq1 2142	. . . . . . . 8 |- ( x = (/) -> ( x e. { (/) } <-> (/) e. { (/) } ) )
6	5	anbi1d 453	. . . . . . 7 |- ( x = (/) -> ( ( x e. { (/) } /\ ph ) <-> ( (/) e. { (/) } /\ ph ) ) )
7	4, 6	bibi12d 233	. . . . . 6 |- ( x = (/) -> ( ( x e. a <-> ( x e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) ) )
8	1, 7	spcv 2692	. . . . 5 |- ( A. x ( x e. a <-> ( x e. { (/) } /\ ph ) ) -> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) )
9	3, 8	eximii 1534	. . . 4 |- E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) )
10	1	snid 3427	. . . . . . . 8 |- (/) e. { (/) }
11	10	biantrur 297	. . . . . . 7 |- ( ph <-> ( (/) e. { (/) } /\ ph ) )
12	11	bicomi 130	. . . . . 6 |- ( ( (/) e. { (/) } /\ ph ) <-> ph )
13	12	bibi2i 225	. . . . 5 |- ( ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ph ) )
14	13	exbii 1537	. . . 4 |- ( E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> E. a ( (/) e. a <-> ph ) )
15	9, 14	mpbi 143	. . 3 |- E. a ( (/) e. a <-> ph )
16		bj-bd0el 10802	. . . . 5 |- BOUNDED (/) e. a
17	16	ax-bj-d0cl 10858	. . . 4 |- DECID (/) e. a
18		bj-dcbi 10862	. . . 4 |- ( ( (/) e. a <-> ph ) -> (DECID (/) e. a <-> DECID ph ) )
19	17, 18	mpbii 146	. . 3 |- ( ( (/) e. a <-> ph ) -> DECID ph )
20	15, 19	eximii 1534	. 2 |- E. aDECID ph
21		bj-ex 10709	. 2 |- ( E. aDECID ph -> DECID ph )
22	20, 21	ax-mp 7	1 |- DECID ph

Syntax hints:   /\ wa 102   <-> wb 103  DECID wdc 776   A.wal 1283   = wceq 1285   E.wex 1422   e. wcel 1434   (/)c0 3252   {csn 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-nul 3906  ax-pr 3966  ax-bd0 10747  ax-bdim 10748  ax-bdor 10750  ax-bdn 10751  ax-bdal 10752  ax-bdex 10753  ax-bdeq 10754  ax-bdsep 10818  ax-bj-d0cl 10858

This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-sn 3406  df-pr 3407  df-bdc 10775

This theorem is referenced by: (None)