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Theorem moeq3 3280
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 2482 . . . . . . 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 1298 . . . . 5  |-  ( y  =  A  ->  (
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
65eubidv 2298 . . . 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 3116 . . . . 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 3278 . . . 4  |-  E! x
( ( ph  /\  x  =  y )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
126, 11vtoclg 3171 . . 3  |-  ( A  e.  _V  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
13 eumo 2308 . . 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 3121 . . . . . . . 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 837 . . . . 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 976 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
21 3orass 976 . . . . 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 1695 . . 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 2345 . . 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 972   A.wal 1377    = wceq 1379    e. wcel 1767   E!weu 2275   E*wmo 2276   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  tz7.44lem1  7068
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