Theorem moeq3 3383

Description: "At most one" property of equality (split into 3 cases). (The first two 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 2633	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
2	1	anbi2d 740	. . . . . 6 |- ( y = A -> ( ( ph /\ x = y ) <-> ( ph /\ x = A ) ) )
3		biidd 252	. . . . . 6 |- ( y = A -> ( ( -. ( ph \/ ps ) /\ x = B ) <-> ( -. ( ph \/ ps ) /\ x = B ) ) )
4		biidd 252	. . . . . 6 |- ( y = A -> ( ( ps /\ x = C ) <-> ( ps /\ x = C ) ) )
5	2, 3, 4	3orbi123d 1398	. . . . 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 2490	. . . 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 3203	. . . . 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 3381	. . . 4 |- E! x ( ( ph /\ x = y ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )
12	6, 11	vtoclg 3266	. . 3 |- ( A e. _V -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
13		eumo 2499	. . 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 17	. 2 |- ( A e. _V -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
15		eqvisset 3211	. . . . . . . 8 |- ( x = A -> A e. _V )
16		pm2.21 120	. . . . . . . 8 |- ( -. A e. _V -> ( A e. _V -> x = y ) )
17	15, 16	syl5 34	. . . . . . 7 |- ( -. A e. _V -> ( x = A -> x = y ) )
18	17	anim2d 589	. . . . . 6 |- ( -. A e. _V -> ( ( ph /\ x = A ) -> ( ph /\ x = y ) ) )
19	18	orim1d 884	. . . . 5 |- ( -. A e. _V -> ( ( ( ph /\ x = A ) \/ ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) -> ( ( ph /\ x = y ) \/ ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) ) )
20		3orass 1040	. . . . 5 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ph /\ x = A ) \/ ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
21		3orass 1040	. . . . 5 |- ( ( ( ph /\ x = y ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ph /\ x = y ) \/ ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
22	19, 20, 21	3imtr4g 285	. . . 4 |- ( -. A e. _V -> ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) -> ( ( ph /\ x = y ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
23	22	alrimiv 1855	. . 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 ) ) ) )
24		euimmo 2522	. . 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 ) ) ) )
25	23, 11, 24	mpisyl 21	. 2 |- ( -. A e. _V -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
26	14, 25	pm2.61i 176	1 |- E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   A.wal 1481   = wceq 1483   e. wcel 1990   E!weu 2470   E*wmo 2471   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  tz7.44lem1  7501