Theorem moeq3 2432

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 1893	. . . . . . 7 |- (y = A -> (x = y <-> x = A))
2	1	anbi2d 678	. . . . . 6 |- (y = A -> ((ph /\ x = y) <-> (ph /\ x = A)))
3		biidd 188	. . . . . 6 |- (y = A -> ((-. (ph \/ ps) /\ x = B) <-> (-. (ph \/ ps) /\ x = B)))
4		biidd 188	. . . . . 6 |- (y = A -> ((ps /\ x = C) <-> (ps /\ x = C)))
5	2, 3, 4	3orbi123d 1167	. . . . 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 1779	. . . 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 2295	. . . . 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 2430	. . . 4 |- E!x((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))
12	6, 11	vtoclg 2346	. . 3 |- (A e. _V -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
13		eumo 1807	. . 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 12	. 2 |- (A e. _V -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
15		pm2.21 92	. . . . . . . . 9 |- (-. A e. _V -> (A e. _V -> x = y))
16		visset 2295	. . . . . . . . . 10 |- x e. _V
17		eleq1 1957	. . . . . . . . . 10 |- (x = A -> (x e. _V <-> A e. _V))
18	16, 17	mpbii 210	. . . . . . . . 9 |- (x = A -> A e. _V)
19	15, 18	syl5 20	. . . . . . . 8 |- (-. A e. _V -> (x = A -> x = y))
20	19	anim2d 620	. . . . . . 7 |- (-. A e. _V -> ((ph /\ x = A) -> (ph /\ x = y)))
21	20	orim1d 625	. . . . . 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 861	. . . . . 6 |- (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((ph /\ x = A) \/ ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
23		3orass 861	. . . . . 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 612	. . . . 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 1664	. . . 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 1816	. . . 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 12	. . 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 55	. 2 |- (-. A e. _V -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
29	14, 28	pm2.61i 140	1 |- E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   \/ w3o 857  A.wal 1296   = wceq 1298   e. wcel 1300  E!weu 1771  E*wmo 1772  _Vcvv 2292

This theorem is referenced by:  tz7.44lem1 5135

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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