Theorem moeq3 1893

Description: "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.)

Hypotheses

Ref	Expression
moeq3.1	|- B e. V
moeq3.2	|- C e. V
moeq3.3	|- -. (ph /\ ps)

Assertion

Ref	Expression
moeq3	|- E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))

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

Proof of Theorem moeq3

Step	Hyp	Ref	Expression
1		eqeq2 1460	. . . . . . 7 |- (y = A -> (x = y <-> x = A))
2	1	anbi2d 614	. . . . . 6 |- (y = A -> ((ph /\ x = y) <-> (ph /\ x = A)))
3		pm4.2i 171	. . . . . 6 |- (y = A -> ((-. (ph \/ ps) /\ x = B) <-> (-. (ph \/ ps) /\ x = B)))
4		pm4.2i 171	. . . . . 6 |- (y = A -> ((ps /\ x = C) <-> (ps /\ x = C)))
5	2, 3, 4	3orbi123d 888	. . . . 5 |- (y = A -> (((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
6	5	eubidv 1363	. . . 4 |- (y = A -> (E!x((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
7		visset 1788	. . . . 5 |- y e. V
8		moeq3.1	. . . . 5 |- B e. V
9		moeq3.2	. . . . 5 |- C e. V
10		moeq3.3	. . . . 5 |- -. (ph /\ ps)
11	7, 8, 9, 10	eueq3 1891	. . . 4 |- E!x((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))
12	6, 11	vtoclg 1822	. . 3 |- (A e. V -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
13		eumo 1388	. . 3 |- (E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
14	12, 13	syl 10	. 2 |- (A e. V -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
15		pm2.21 76	. . . . . . . . 9 |- (-. A e. V -> (A e. V -> x = y))
16		visset 1788	. . . . . . . . . 10 |- x e. V
17		eleq1 1510	. . . . . . . . . 10 |- (x = A -> (x e. V <-> A e. V))
18	16, 17	mpbii 193	. . . . . . . . 9 |- (x = A -> A e. V)
19	15, 18	syl5 21	. . . . . . . 8 |- (-. A e. V -> (x = A -> x = y))
20	19	anim2d 559	. . . . . . 7 |- (-. A e. V -> ((ph /\ x = A) -> (ph /\ x = y)))
21	20	orim1d 564	. . . . . 6 |- (-. A e. V -> (((ph /\ x = A) \/ ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))) -> ((ph /\ x = y) \/ ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))))
22		3orass 775	. . . . . 6 |- (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((ph /\ x = A) \/ ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
23		3orass 775	. . . . . 6 |- (((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((ph /\ x = y) \/ ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
24	21, 22, 23	3imtr4g 551	. . . . 5 |- (-. A e. V -> (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) -> ((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
25	24	19.21aiv 1268	. . . 4 |- (-. A e. V -> A.x(((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) -> ((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
26		euimmo 1397	. . . 4 |- (A.x(((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) -> ((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))) -> (E!x((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
27	25, 26	syl 10	. . 3 |- (-. A e. V -> (E!x((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
28	11, 27	mpi 44	. 2 |- (-. A e. V -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
29	14, 28	pm2.61i 126	1 |- E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   \/ w3o 771  A.wal 950   = wceq 1099   e. wcel 1105  E!weu 1357  E*wmo 1358  Vcvv 1786

This theorem is referenced by:  tz7.44lem1 3866

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-3or 773  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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