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Theorem moeq3 3111
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 2445 . . . . . . 7  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
21anbi2d 685 . . . . . 6  |-  ( y  =  A  ->  (
( ph  /\  x  =  y )  <->  ( ph  /\  x  =  A ) ) )
3 biidd 229 . . . . . 6  |-  ( y  =  A  ->  (
( -.  ( ph  \/  ps )  /\  x  =  B )  <->  ( -.  ( ph  \/  ps )  /\  x  =  B
) ) )
4 biidd 229 . . . . . 6  |-  ( y  =  A  ->  (
( ps  /\  x  =  C )  <->  ( ps  /\  x  =  C ) ) )
52, 3, 43orbi123d 1253 . . . . 5  |-  ( y  =  A  ->  (
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
65eubidv 2289 . . . 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 2959 . . . . 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 3109 . . . 4  |-  E! x
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
126, 11vtoclg 3011 . . 3  |-  ( A  e.  _V  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
13 eumo 2321 . . 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 vex 2959 . . . . . . . . 9  |-  x  e. 
_V
16 eleq1 2496 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
1715, 16mpbii 203 . . . . . . . 8  |-  ( x  =  A  ->  A  e.  _V )
18 pm2.21 102 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  ( A  e.  _V  ->  x  =  y ) )
1917, 18syl5 30 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( x  =  A  ->  x  =  y )
)
2019anim2d 549 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( ( ph  /\  x  =  A )  ->  ( ph  /\  x  =  y ) ) )
2120orim1d 813 . . . . 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 939 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
23 3orass 939 . . . . 5  |-  ( ( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  <->  ( ( ph  /\  x  =  y )  \/  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
2421, 22, 233imtr4g 262 . . . 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 1641 . . 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 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 ) ) ) )
2725, 11, 26ee10 1385 . 2  |-  ( -.  A  e.  _V  ->  E* x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
2814, 27pm2.61i 158 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 358    /\ wa 359    \/ w3o 935   A.wal 1549    = wceq 1652    e. wcel 1725   E!weu 2281   E*wmo 2282   _Vcvv 2956
This theorem is referenced by:  tz7.44lem1  6663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958
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