Theorem eueq2 1429

Description: Equality has existential uniqueness (split into 2 cases).

Hypotheses

Ref	Expression
eueq2.1	|- A e. V
eueq2.2	|- B e. V

Assertion

Ref	Expression
eueq2	|- E!x((ph /\ x = A) \/ (-. ph /\ x = B))

Distinct variable group(s):   ph,x   x,A   x,B

Proof of Theorem eueq2

Step	Hyp	Ref	Expression
1		euorv 1025	. . . 4 |- ((-. -. ph /\ E!x(ph /\ x = A)) -> E!x(-. ph \/ (ph /\ x = A)))
2		negb 79	. . . 4 |- (ph -> -. -. ph)
3		eueq2.1	. . . . . 6 |- A e. V
4	3	eueq1 1428	. . . . 5 |- E!x x = A
5		euanv 1053	. . . . . 6 |- (E!x(ph /\ x = A) <-> (ph /\ E!x x = A))
6	5	biimpr 134	. . . . 5 |- ((ph /\ E!x x = A) -> E!x(ph /\ x = A))
7	4, 6	mpan2 519	. . . 4 |- (ph -> E!x(ph /\ x = A))
8	1, 2, 7	sylanc 361	. . 3 |- (ph -> E!x(-. ph \/ (ph /\ x = A)))
9	2	bianfd 554	. . . . . 6 |- (ph -> (-. ph <-> (-. ph /\ x = B)))
10	9	orbi2d 466	. . . . 5 |- (ph -> (((ph /\ x = A) \/ -. ph) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
11		orcom 209	. . . . 5 |- ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ -. ph))
12	10, 11	syl5bb 410	. . . 4 |- (ph -> ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
13	12	bieudv 1013	. . 3 |- (ph -> (E!x(-. ph \/ (ph /\ x = A)) <-> E!x((ph /\ x = A) \/ (-. ph /\ x = B))))
14	8, 13	mpbid 170	. 2 |- (ph -> E!x((ph /\ x = A) \/ (-. ph /\ x = B)))
15		eueq2.2	. . . . . 6 |- B e. V
16	15	eueq1 1428	. . . . 5 |- E!x x = B
17		euanv 1053	. . . . . 6 |- (E!x(-. ph /\ x = B) <-> (-. ph /\ E!x x = B))
18	17	biimpr 134	. . . . 5 |- ((-. ph /\ E!x x = B) -> E!x(-. ph /\ x = B))
19	16, 18	mpan2 519	. . . 4 |- (-. ph -> E!x(-. ph /\ x = B))
20		euorv 1025	. . . 4 |- ((-. ph /\ E!x(-. ph /\ x = B)) -> E!x(ph \/ (-. ph /\ x = B)))
21	19, 20	mpdan 527	. . 3 |- (-. ph -> E!x(ph \/ (-. ph /\ x = B)))
22		id 9	. . . . . 6 |- (-. ph -> -. ph)
23	22	bianfd 554	. . . . 5 |- (-. ph -> (ph <-> (ph /\ x = A)))
24	23	orbi1d 467	. . . 4 |- (-. ph -> ((ph \/ (-. ph /\ x = B)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
25	24	bieudv 1013	. . 3 |- (-. ph -> (E!x(ph \/ (-. ph /\ x = B)) <-> E!x((ph /\ x = A) \/ (-. ph /\ x = B))))
26	21, 25	mpbid 170	. 2 |- (-. ph -> E!x((ph /\ x = A) \/ (-. ph /\ x = B)))
27	14, 26	pm2.61i 110	1 |- E!x((ph /\ x = A) \/ (-. ph /\ x = B))

Syntax hints:  -. wn 1   \/ wo 195   /\ wa 196  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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