Theorem eueq2dc 2899

Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)

Hypotheses

Ref	Expression
eueq2dc.1	|- A e. _V
eueq2dc.2	|- B e. _V

Assertion

Ref	Expression
eueq2dc	|- (DECID ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )

Distinct variable groups:   ph, x   x, A   x, B

Proof of Theorem eueq2dc

Step	Hyp	Ref	Expression
1		df-dc 825	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 619	. . . . 5 |- ( ph -> -. -. ph )
3		eueq2dc.1	. . . . . . 7 |- A e. _V
4	3	eueq1 2898	. . . . . 6 |- E! x x = A
5		euanv 2071	. . . . . . 7 |- ( E! x ( ph /\ x = A ) <-> ( ph /\ E! x x = A ) )
6	5	biimpri 132	. . . . . 6 |- ( ( ph /\ E! x x = A ) -> E! x ( ph /\ x = A ) )
7	4, 6	mpan2 422	. . . . 5 |- ( ph -> E! x ( ph /\ x = A ) )
8		euorv 2041	. . . . 5 |- ( ( -. -. ph /\ E! x ( ph /\ x = A ) ) -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
9	2, 7, 8	syl2anc 409	. . . 4 |- ( ph -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
10		orcom 718	. . . . . 6 |- ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ -. ph ) )
11	2	bianfd 938	. . . . . . 7 |- ( ph -> ( -. ph <-> ( -. ph /\ x = B ) ) )
12	11	orbi2d 780	. . . . . 6 |- ( ph -> ( ( ( ph /\ x = A ) \/ -. ph ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
13	10, 12	syl5bb 191	. . . . 5 |- ( ph -> ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
14	13	eubidv 2022	. . . 4 |- ( ph -> ( E! x ( -. ph \/ ( ph /\ x = A ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
15	9, 14	mpbid 146	. . 3 |- ( ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
16		eueq2dc.2	. . . . . . 7 |- B e. _V
17	16	eueq1 2898	. . . . . 6 |- E! x x = B
18		euanv 2071	. . . . . . 7 |- ( E! x ( -. ph /\ x = B ) <-> ( -. ph /\ E! x x = B ) )
19	18	biimpri 132	. . . . . 6 |- ( ( -. ph /\ E! x x = B ) -> E! x ( -. ph /\ x = B ) )
20	17, 19	mpan2 422	. . . . 5 |- ( -. ph -> E! x ( -. ph /\ x = B ) )
21		euorv 2041	. . . . 5 |- ( ( -. ph /\ E! x ( -. ph /\ x = B ) ) -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
22	20, 21	mpdan 418	. . . 4 |- ( -. ph -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
23		id 19	. . . . . . 7 |- ( -. ph -> -. ph )
24	23	bianfd 938	. . . . . 6 |- ( -. ph -> ( ph <-> ( ph /\ x = A ) ) )
25	24	orbi1d 781	. . . . 5 |- ( -. ph -> ( ( ph \/ ( -. ph /\ x = B ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
26	25	eubidv 2022	. . . 4 |- ( -. ph -> ( E! x ( ph \/ ( -. ph /\ x = B ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
27	22, 26	mpbid 146	. . 3 |- ( -. ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
28	15, 27	jaoi 706	. 2 |- ( ( ph \/ -. ph ) -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
29	1, 28	sylbi 120	1 |- (DECID ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   = wceq 1343   E!weu 2014   e. wcel 2136   _Vcvv 2726

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-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by: (None)