Theorem bj-d0clsepcl 16868

Description: Δ₀-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) New usage is discouraged since this statement is not intuitionnistic. (New usage 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 4258	. . . . . . 7 |- (/) e. _V
2	1	bj-snex 16856	. . . . . 6 |- { (/) } e. _V
3	2	sepgi 4250	. . . . 5 |- E. a A. x ( x e. a <-> ( x e. { (/) } /\ ph ) )
4		eleq1 2301	. . . . . . 7 |- ( x = (/) -> ( x e. a <-> (/) e. a ) )
5		eleq1 2301	. . . . . . . 8 |- ( x = (/) -> ( x e. { (/) } <-> (/) e. { (/) } ) )
6	5	anbi1d 469	. . . . . . 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 2919	. . . . 5 |- ( A. x ( x e. a <-> ( x e. { (/) } /\ ph ) ) -> ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) ) )
9	3, 8	eximii 1655	. . . 4 |- E. a ( (/) e. a <-> ( (/) e. { (/) } /\ ph ) )
10	1	snid 3739	. . . . . . . 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 1658	. . . 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 16811	. . . . 5 |- BOUNDED (/) e. a
17	16	ax-bj-d0cl 16867	. . . 4 |- DECID (/) e. a
18		dcbiit 851	. . . 4 |- ( ( (/) e. a <-> ph ) -> (DECID (/) e. a <-> DECID ph ) )
19	17, 18	mpbii 148	. . 3 |- ( ( (/) e. a <-> ph ) -> DECID ph )
20	15, 19	eximii 1655	. 2 |- E. aDECID ph
21		bj-ex 16707	. 2 |- ( E. aDECID ph -> DECID ph )
22	20, 21	ax-mp 5	1 |- DECID ph

Syntax hints:   /\ wa 104   <-> wb 105  DECID wdc 846   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   (/)c0 3520   {csn 3708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pr 4344  ax-bd0 16756  ax-bdim 16757  ax-bdor 16759  ax-bdn 16760  ax-bdal 16761  ax-bdex 16762  ax-bdeq 16763  ax-bdsep 16827  ax-bj-d0cl 16867

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3714  df-pr 3715  df-bdc 16784

This theorem is referenced by: (None)