Theorem bj-d0clsepcl 15417

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 4156	. . . . . . 7 |- (/) e. _V
2	1	bj-snex 15405	. . . . . 6 |- { (/) } e. _V
3	2	zfauscl 4149	. . . . 5 |- E. a A. x ( x e. a <-> ( x e. { (/) } /\ ph ) )
4		eleq1 2256	. . . . . . 7 |- ( x = (/) -> ( x e. a <-> (/) e. a ) )
5		eleq1 2256	. . . . . . . 8 |- ( x = (/) -> ( x e. { (/) } <-> (/) e. { (/) } ) )
6	5	anbi1d 465	. . . . . . 7 |- ( x = (/) -> ( ( x e. { (/) } /\ ph ) <-> ( (/) e. { (/) } /\ ph ) ) )
7	4, 6	bibi12d 235	. . . . . 6 |- ( x = (/) -> ( ( x e. a <-> ( x e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) ) )
8	1, 7	spcv 2854	. . . . 5 |- ( A. x ( x e. a <-> ( x e. { (/) } /\ ph ) ) -> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) )
9	3, 8	eximii 1613	. . . 4 |- E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) )
10	1	snid 3649	. . . . . . . 8 |- (/) e. { (/) }
11	10	biantrur 303	. . . . . . 7 |- ( ph <-> ( (/) e. { (/) } /\ ph ) )
12	11	bicomi 132	. . . . . 6 |- ( ( (/) e. { (/) } /\ ph ) <-> ph )
13	12	bibi2i 227	. . . . 5 |- ( ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> ( (/) e. a <-> ph ) )
14	13	exbii 1616	. . . 4 |- ( E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) <-> E. a ( (/) e. a <-> ph ) )
15	9, 14	mpbi 145	. . 3 |- E. a ( (/) e. a <-> ph )
16		bj-bd0el 15360	. . . . 5 |- BOUNDED (/) e. a
17	16	ax-bj-d0cl 15416	. . . 4 |- DECID (/) e. a
18		dcbiit 840	. . . 4 |- ( ( (/) e. a <-> ph ) -> (DECID (/) e. a <-> DECID ph ) )
19	17, 18	mpbii 148	. . 3 |- ( ( (/) e. a <-> ph ) -> DECID ph )
20	15, 19	eximii 1613	. 2 |- E. aDECID ph
21		bj-ex 15254	. 2 |- ( E. aDECID ph -> DECID ph )
22	20, 21	ax-mp 5	1 |- DECID ph

Syntax hints:   /\ wa 104   <-> wb 105  DECID wdc 835   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   (/)c0 3446   {csn 3618

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pr 4238  ax-bd0 15305  ax-bdim 15306  ax-bdor 15308  ax-bdn 15309  ax-bdal 15310  ax-bdex 15311  ax-bdeq 15312  ax-bdsep 15376  ax-bj-d0cl 15416

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-bdc 15333

This theorem is referenced by: (None)