Theorem moeq3 3014

Description: "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)

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

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . . . . . 7 |- (y = A -> (x = y <-> x = A))
2	1	anbi2d 684	. . . . . 6 |- (y = A -> ((ph /\ x = y) <-> (ph /\ x = A)))
3		biidd 228	. . . . . 6 |- (y = A -> ((-. (ph \/ ps) /\ x = B) <-> (-. (ph \/ ps) /\ x = B)))
4		biidd 228	. . . . . 6 |- (y = A -> ((ps /\ x = C) <-> (ps /\ x = C)))
5	2, 3, 4	3orbi123d 1251	. . . . 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 2212	. . . 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		vex 2863	. . . . 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 3012	. . . 4 |- E!x((ph /\ x = y) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))
12	6, 11	vtoclg 2915	. . 3 |- (A e. _V -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
13		eumo 2244	. . 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 15	. 2 |- (A e. _V -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
15		vex 2863	. . . . . . . . 9 |- x e. _V
16		eleq1 2413	. . . . . . . . 9 |- (x = A -> (x e. _V <-> A e. _V))
17	15, 16	mpbii 202	. . . . . . . 8 |- (x = A -> A e. _V)
18		pm2.21 100	. . . . . . . 8 |- (-. A e. _V -> (A e. _V -> x = y))
19	17, 18	syl5 28	. . . . . . 7 |- (-. A e. _V -> (x = A -> x = y))
20	19	anim2d 548	. . . . . 6 |- (-. A e. _V -> ((ph /\ x = A) -> (ph /\ x = y)))
21	20	orim1d 812	. . . . 5 |- (-. 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 937	. . . . 5 |- (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((ph /\ x = A) \/ ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
23		3orass 937	. . . . 5 |- (((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 261	. . . 4 |- (-. 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	alrimiv 1631	. . 3 |- (-. 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 2253	. . 3 |- (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, 11, 26	ee10 1376	. 2 |- (-. A e. _V -> E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
28	14, 27	pm2.61i 156	1 |- E*x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358   \/ w3o 933  A.wal 1540   = wceq 1642   e. wcel 1710  E!weu 2204  E*wmo 2205  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by: (None)