Theorem eueq2 1890

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 groups:   ph,x   x,A   x,B

Proof of Theorem eueq2

Step	Hyp	Ref	Expression
1		euorv 1376	. . . 4 |- ((-. -. ph /\ E!x(ph /\ x = A)) -> E!x(-. ph \/ (ph /\ x = A)))
2		negb 86	. . . 4 |- (ph -> -. -. ph)
3		eueq2.1	. . . . . 6 |- A e. V
4	3	eueq1 1889	. . . . 5 |- E!x x = A
5		euanv 1409	. . . . . 6 |- (E!x(ph /\ x = A) <-> (ph /\ E!x x = A))
6	5	biimpr 152	. . . . 5 |- ((ph /\ E!x x = A) -> E!x(ph /\ x = A))
7	4, 6	mpan2 693	. . . 4 |- (ph -> E!x(ph /\ x = A))
8	1, 2, 7	sylanc 471	. . 3 |- (ph -> E!x(-. ph \/ (ph /\ x = A)))
9	2	bianfd 735	. . . . . 6 |- (ph -> (-. ph <-> (-. ph /\ x = B)))
10	9	orbi2d 612	. . . . 5 |- (ph -> (((ph /\ x = A) \/ -. ph) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
11		orcom 246	. . . . 5 |- ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ -. ph))
12	10, 11	syl5bb 530	. . . 4 |- (ph -> ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
13	12	eubidv 1363	. . 3 |- (ph -> (E!x(-. ph \/ (ph /\ x = A)) <-> E!x((ph /\ x = A) \/ (-. ph /\ x = B))))
14	8, 13	mpbid 195	. 2 |- (ph -> E!x((ph /\ x = A) \/ (-. ph /\ x = B)))
15		eueq2.2	. . . . . 6 |- B e. V
16	15	eueq1 1889	. . . . 5 |- E!x x = B
17		euanv 1409	. . . . . 6 |- (E!x(-. ph /\ x = B) <-> (-. ph /\ E!x x = B))
18	17	biimpr 152	. . . . 5 |- ((-. ph /\ E!x x = B) -> E!x(-. ph /\ x = B))
19	16, 18	mpan2 693	. . . 4 |- (-. ph -> E!x(-. ph /\ x = B))
20		euorv 1376	. . . 4 |- ((-. ph /\ E!x(-. ph /\ x = B)) -> E!x(ph \/ (-. ph /\ x = B)))
21	19, 20	mpdan 701	. . 3 |- (-. ph -> E!x(ph \/ (-. ph /\ x = B)))
22		id 59	. . . . . 6 |- (-. ph -> -. ph)
23	22	bianfd 735	. . . . 5 |- (-. ph -> (ph <-> (ph /\ x = A)))
24	23	orbi1d 613	. . . 4 |- (-. ph -> ((ph \/ (-. ph /\ x = B)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
25	24	eubidv 1363	. . 3 |- (-. ph -> (E!x(ph \/ (-. ph /\ x = B)) <-> E!x((ph /\ x = A) \/ (-. ph /\ x = B))))
26	21, 25	mpbid 195	. 2 |- (-. ph -> E!x((ph /\ x = A) \/ (-. ph /\ x = B)))
27	14, 26	pm2.61i 126	1 |- E!x((ph /\ x = A) \/ (-. ph /\ x = B))

Syntax hints:  -. wn 2   \/ wo 222   /\ wa 223   = wceq 1099   e. wcel 1105  E!weu 1357  Vcvv 1786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787

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