Theorem eueq2 1918

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 1399	. . . 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 1917	. . . . 5 |- E!x x = A
5		euanv 1432	. . . . . 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 696	. . . 4 |- (ph -> E!x(ph /\ x = A))
8	1, 2, 7	sylanc 471	. . 3 |- (ph -> E!x(-. ph \/ (ph /\ x = A)))
9	2	bianfd 738	. . . . . 6 |- (ph -> (-. ph <-> (-. ph /\ x = B)))
10	9	orbi2d 614	. . . . 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 532	. . . 4 |- (ph -> ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
13	12	eubidv 1386	. . 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 1917	. . . . 5 |- E!x x = B
17		euanv 1432	. . . . . 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 696	. . . 4 |- (-. ph -> E!x(-. ph /\ x = B))
20		euorv 1399	. . . 4 |- ((-. ph /\ E!x(-. ph /\ x = B)) -> E!x(ph \/ (-. ph /\ x = B)))
21	19, 20	mpdan 704	. . 3 |- (-. ph -> E!x(ph \/ (-. ph /\ x = B)))
22		id 59	. . . . . 6 |- (-. ph -> -. ph)
23	22	bianfd 738	. . . . 5 |- (-. ph -> (ph <-> (ph /\ x = A)))
24	23	orbi1d 615	. . . 4 |- (-. ph -> ((ph \/ (-. ph /\ x = B)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
25	24	eubidv 1386	. . 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 956   e. wcel 958  E!weu 1380  Vcvv 1811

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

Copyright terms: Public domain