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Theorem moeq3 3233
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 2466 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21anbi2d 703 . . . . . 6  |-  ( y  =  A  ->  (
( ph  /\  x  =  y )  <->  ( ph  /\  x  =  A ) ) )
3 biidd 237 . . . . . 6  |-  ( y  =  A  ->  (
( -.  ( ph  \/  ps )  /\  x  =  B )  <->  ( -.  ( ph  \/  ps )  /\  x  =  B
) ) )
4 biidd 237 . . . . . 6  |-  ( y  =  A  ->  (
( ps  /\  x  =  C )  <->  ( ps  /\  x  =  C ) ) )
52, 3, 43orbi123d 1289 . . . . 5  |-  ( y  =  A  ->  (
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
65eubidv 2283 . . . 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 3071 . . . . 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 3231 . . . 4  |-  E! x
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
126, 11vtoclg 3126 . . 3  |-  ( A  e.  _V  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
13 eumo 2293 . . 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 16 . 2  |-  ( A  e.  _V  ->  E* x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
15 eqvisset 3076 . . . . . . . 8  |-  ( x  =  A  ->  A  e.  _V )
16 pm2.21 108 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  ( A  e.  _V  ->  x  =  y ) )
1715, 16syl5 32 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( x  =  A  ->  x  =  y )
)
1817anim2d 565 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( ( ph  /\  x  =  A )  ->  ( ph  /\  x  =  y ) ) )
1918orim1d 835 . . . . 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 968 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
21 3orass 968 . . . . 5  |-  ( ( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  <->  ( ( ph  /\  x  =  y )  \/  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
2219, 20, 213imtr4g 270 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  ->  (
( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
2322alrimiv 1686 . . 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 2330 . . 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 ) ) ) )
2523, 11, 24mpisyl 18 . 2  |-  ( -.  A  e.  _V  ->  E* x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
2614, 25pm2.61i 164 1  |-  E* x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964   A.wal 1368    = wceq 1370    e. wcel 1758   E!weu 2260   E*wmo 2261   _Vcvv 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3070
This theorem is referenced by:  tz7.44lem1  6961
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