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Theorem eueq3dc 2787
Description: Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
Hypotheses
Ref Expression
eueq3dc.1  |-  A  e. 
_V
eueq3dc.2  |-  B  e. 
_V
eueq3dc.3  |-  C  e. 
_V
eueq3dc.4  |-  -.  ( ph  /\  ps )
Assertion
Ref Expression
eueq3dc  |-  (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 eueq3dc
StepHypRef Expression
1 dcor 881 . 2  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
2 df-dc 781 . . 3  |-  (DECID  ( ph  \/  ps )  <->  ( ( ph  \/  ps )  \/ 
-.  ( ph  \/  ps ) ) )
3 eueq3dc.1 . . . . . . 7  |-  A  e. 
_V
43eueq1 2785 . . . . . 6  |-  E! x  x  =  A
5 ibar 295 . . . . . . . . 9  |-  ( ph  ->  ( x  =  A  <-> 
( ph  /\  x  =  A ) ) )
6 pm2.45 692 . . . . . . . . . . . . 13  |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
7 eueq3dc.4 . . . . . . . . . . . . . . 15  |-  -.  ( ph  /\  ps )
87imnani 660 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ps )
98con2i 592 . . . . . . . . . . . . 13  |-  ( ps 
->  -.  ph )
106, 9jaoi 671 . . . . . . . . . . . 12  |-  ( ( -.  ( ph  \/  ps )  \/  ps )  ->  -.  ph )
1110con2i 592 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( -.  ( ph  \/  ps )  \/ 
ps ) )
126con2i 592 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  -.  ( ph  \/  ps ) )
1312bianfd 894 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ( ph  \/  ps )  <->  ( -.  ( ph  \/  ps )  /\  x  =  B
) ) )
148bianfd 894 . . . . . . . . . . . 12  |-  ( ph  ->  ( ps  <->  ( ps  /\  x  =  C ) ) )
1513, 14orbi12d 742 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -.  ( ph  \/  ps )  \/ 
ps )  <->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
1611, 15mtbid 632 . . . . . . . . . 10  |-  ( ph  ->  -.  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
17 biorf 698 . . . . . . . . . 10  |-  ( -.  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  -> 
( ( ph  /\  x  =  A )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
1816, 17syl 14 . . . . . . . . 9  |-  ( ph  ->  ( ( ph  /\  x  =  A )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
195, 18bitrd 186 . . . . . . . 8  |-  ( ph  ->  ( x  =  A  <-> 
( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
20 3orrot 930 . . . . . . . . 9  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C )  \/  ( ph  /\  x  =  A ) ) )
21 df-3or 925 . . . . . . . . 9  |-  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C )  \/  ( ph  /\  x  =  A ) )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) )
2220, 21bitri 182 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) )
2319, 22syl6bbr 196 . . . . . . 7  |-  ( ph  ->  ( x  =  A  <-> 
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
2423eubidv 1956 . . . . . 6  |-  ( ph  ->  ( E! x  x  =  A  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
254, 24mpbii 146 . . . . 5  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
26 eueq3dc.3 . . . . . . 7  |-  C  e. 
_V
2726eueq1 2785 . . . . . 6  |-  E! x  x  =  C
28 ibar 295 . . . . . . . . 9  |-  ( ps 
->  ( x  =  C  <-> 
( ps  /\  x  =  C ) ) )
298adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  =  A )  ->  -.  ps )
30 pm2.46 693 . . . . . . . . . . . . 13  |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
3130adantr 270 . . . . . . . . . . . 12  |-  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  ->  -.  ps )
3229, 31jaoi 671 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  ->  -.  ps )
3332con2i 592 . . . . . . . . . 10  |-  ( ps 
->  -.  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )
) )
34 biorf 698 . . . . . . . . . 10  |-  ( -.  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  -> 
( ( ps  /\  x  =  C )  <->  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) ) )
3533, 34syl 14 . . . . . . . . 9  |-  ( ps 
->  ( ( ps  /\  x  =  C )  <->  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) ) )
3628, 35bitrd 186 . . . . . . . 8  |-  ( ps 
->  ( x  =  C  <-> 
( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )
)  \/  ( ps 
/\  x  =  C ) ) ) )
37 df-3or 925 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) )
3836, 37syl6bbr 196 . . . . . . 7  |-  ( ps 
->  ( x  =  C  <-> 
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
3938eubidv 1956 . . . . . 6  |-  ( ps 
->  ( E! x  x  =  C  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
4027, 39mpbii 146 . . . . 5  |-  ( ps 
->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
4125, 40jaoi 671 . . . 4  |-  ( (
ph  \/  ps )  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
42 eueq3dc.2 . . . . . 6  |-  B  e. 
_V
4342eueq1 2785 . . . . 5  |-  E! x  x  =  B
44 ibar 295 . . . . . . . 8  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( -.  ( ph  \/  ps )  /\  x  =  B )
) )
45 simpl 107 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  ph )
46 simpl 107 . . . . . . . . . . 11  |-  ( ( ps  /\  x  =  C )  ->  ps )
4745, 46orim12i 711 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  ->  ( ph  \/  ps ) )
4847con3i 597 . . . . . . . . 9  |-  ( -.  ( ph  \/  ps )  ->  -.  ( ( ph  /\  x  =  A )  \/  ( ps 
/\  x  =  C ) ) )
49 biorf 698 . . . . . . . . 9  |-  ( -.  ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  -> 
( ( -.  ( ph  \/  ps )  /\  x  =  B )  <->  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
5048, 49syl 14 . . . . . . . 8  |-  ( -.  ( ph  \/  ps )  ->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  <->  ( ( (
ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
5144, 50bitrd 186 . . . . . . 7  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( ( (
ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
52 3orcomb 933 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( ps 
/\  x  =  C )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
53 df-3or 925 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
5452, 53bitri 182 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
5551, 54syl6bbr 196 . . . . . 6  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
5655eubidv 1956 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  ( E! x  x  =  B  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
5743, 56mpbii 146 . . . 4  |-  ( -.  ( ph  \/  ps )  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
5841, 57jaoi 671 . . 3  |-  ( ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps ) )  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
592, 58sylbi 119 . 2  |-  (DECID  ( ph  \/  ps )  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) )
601, 59syl6 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 102    <-> wb 103    \/ wo 664  DECID wdc 780    \/ w3o 923    = wceq 1289    e. wcel 1438   E!weu 1948   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  moeq3dc  2789
  Copyright terms: Public domain W3C validator