ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  moeq3dc Unicode version

Theorem moeq3dc 2888
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 2886 . 2  |-  (DECID  ph  ->  (DECID  ps 
->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
6 eumo 2038 . 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 698  DECID wdc 820    \/ w3o 962    = wceq 1335   E!weu 2006   E*wmo 2007    e. wcel 2128   _Vcvv 2712
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator