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Theorem moeq3 2955
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.
StepHypRef Expression
1 eqeq2 2305 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21anbi2d 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 ) ) )
52, 3, 43orbi123d 1251 . . . . 5  |-  ( y  =  A  ->  (
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
65eubidv 2164 . . . 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 2804 . . . . 5  |-  y  e. 
_V
8 moeq3.1 . . . . 5  |-  B  e. 
_V
9 moeq3.2 . . . . 5  |-  C  e. 
_V
10 moeq3.3 . . . . 5  |-  -.  ( ph  /\  ps )
117, 8, 9, 10eueq3 2953 . . . 4  |-  E! x
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
126, 11vtoclg 2856 . . 3  |-  ( A  e.  _V  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
13 eumo 2196 . . 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 ) ) )
1412, 13syl 15 . 2  |-  ( A  e.  _V  ->  E* x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
15 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
16 eleq1 2356 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
1715, 16mpbii 202 . . . . . . . 8  |-  ( x  =  A  ->  A  e.  _V )
18 pm2.21 100 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  ( A  e.  _V  ->  x  =  y ) )
1917, 18syl5 28 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( x  =  A  ->  x  =  y )
)
2019anim2d 548 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( ( ph  /\  x  =  A )  ->  ( ph  /\  x  =  y ) ) )
2120orim1d 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 ) ) ) )
2421, 22, 233imtr4g 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 ) ) ) )
2524alrimiv 1621 . . 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 2205 . . 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 ) ) ) )
2725, 11, 26ee10 1366 . 2  |-  ( -.  A  e.  _V  ->  E* x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
2814, 27pm2.61i 156 1  |-  E* x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    \/ w3o 933   A.wal 1530    = wceq 1632    e. wcel 1696   E!weu 2156   E*wmo 2157   _Vcvv 2801
This theorem is referenced by:  tz7.44lem1  6434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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