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Theorem moeq3dc 2833
Description: "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
Hypotheses
Ref Expression
moeq3dc.1  |-  A  e. 
_V
moeq3dc.2  |-  B  e. 
_V
moeq3dc.3  |-  C  e. 
_V
moeq3dc.4  |-  -.  ( ph  /\  ps )
Assertion
Ref Expression
moeq3dc  |-  (DECID  ph  ->  (DECID  ps 
->  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 moeq3dc
StepHypRef Expression
1 moeq3dc.1 . . 3  |-  A  e. 
_V
2 moeq3dc.2 . . 3  |-  B  e. 
_V
3 moeq3dc.3 . . 3  |-  C  e. 
_V
4 moeq3dc.4 . . 3  |-  -.  ( ph  /\  ps )
51, 2, 3, 4eueq3dc 2831 . 2  |-  (DECID  ph  ->  (DECID  ps 
->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
6 eumo 2009 . 2  |-  ( E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  ->  E* x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
75, 6syl6 33 1  |-  (DECID  ph  ->  (DECID  ps 
->  E* x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 682  DECID wdc 804    \/ w3o 946    = wceq 1316    e. wcel 1465   E!weu 1977   E*wmo 1978   _Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by: (None)
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