Metamath Proof Explorer

Theorem moeq3

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)

Ref Expression
Hypotheses moeq3.1 
|- B e. _V
moeq3.2 
|- C e. _V
moeq3.3 
|- -. ( ph /\ ps )
Assertion moeq3 
|- E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )

Proof

Step Hyp Ref Expression
1 moeq3.1 
 |-  B e. _V
2 moeq3.2 
 |-  C e. _V
3 moeq3.3 
 |-  -. ( ph /\ ps )
4 eqeq2 
 |-  ( y = A -> ( x = y <-> x = A ) )
5 4 anbi2d 
 |-  ( y = A -> ( ( ph /\ x = y ) <-> ( ph /\ x = A ) ) )
6 biidd 
 |-  ( y = A -> ( ( -. ( ph \/ ps ) /\ x = B ) <-> ( -. ( ph \/ ps ) /\ x = B ) ) )
7 biidd 
 |-  ( y = A -> ( ( ps /\ x = C ) <-> ( ps /\ x = C ) ) )
8 5 6 7 3orbi123d 
 |-  ( y = A -> ( ( ( ph /\ x = y ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
9 8 eubidv 
 |-  ( 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 ) ) ) )
10 vex 
 |-  y e. _V
11 10 1 2 3 eueq3 
 |-  E! x ( ( ph /\ x = y ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )
12 9 11 vtoclg 
 |-  ( A e. _V -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
13 eumo 
 |-  ( 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 
 |-  ( A e. _V -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
15 eqvisset 
 |-  ( x = A -> A e. _V )
16 pm2.21 
 |-  ( -. A e. _V -> ( A e. _V -> x = y ) )
17 15 16 syl5 
 |-  ( -. A e. _V -> ( x = A -> x = y ) )
18 17 anim2d 
 |-  ( -. A e. _V -> ( ( ph /\ x = A ) -> ( ph /\ x = y ) ) )
19 18 orim1d 
 |-  ( -. 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 
 |-  ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ph /\ x = A ) \/ ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
21 3orass 
 |-  ( ( ( 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 
 |-  ( -. 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 
 |-  ( -. 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 
 |-  ( 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 
 |-  ( -. A e. _V -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
26 14 25 pm2.61i 
 |-  E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) )