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Theorem eueq3dc 2829
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 902 . 2  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
2 df-dc 803 . . 3  |-  (DECID  ( ph  \/  ps )  <->  ( ( ph  \/  ps )  \/ 
-.  ( ph  \/  ps ) ) )
3 eueq3dc.1 . . . . . . 7  |-  A  e. 
_V
43eueq1 2827 . . . . . 6  |-  E! x  x  =  A
5 ibar 297 . . . . . . . . 9  |-  ( ph  ->  ( x  =  A  <-> 
( ph  /\  x  =  A ) ) )
6 pm2.45 710 . . . . . . . . . . . . 13  |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
7 eueq3dc.4 . . . . . . . . . . . . . . 15  |-  -.  ( ph  /\  ps )
87imnani 663 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ps )
98con2i 599 . . . . . . . . . . . . 13  |-  ( ps 
->  -.  ph )
106, 9jaoi 688 . . . . . . . . . . . 12  |-  ( ( -.  ( ph  \/  ps )  \/  ps )  ->  -.  ph )
1110con2i 599 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( -.  ( ph  \/  ps )  \/ 
ps ) )
126con2i 599 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  -.  ( ph  \/  ps ) )
1312bianfd 915 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ( ph  \/  ps )  <->  ( -.  ( ph  \/  ps )  /\  x  =  B
) ) )
148bianfd 915 . . . . . . . . . . . 12  |-  ( ph  ->  ( ps  <->  ( ps  /\  x  =  C ) ) )
1513, 14orbi12d 765 . . . . . . . . . . 11  |-  ( ph  ->  ( ( -.  ( ph  \/  ps )  \/ 
ps )  <->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
1611, 15mtbid 644 . . . . . . . . . 10  |-  ( ph  ->  -.  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
17 biorf 716 . . . . . . . . . 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 187 . . . . . . . 8  |-  ( ph  ->  ( x  =  A  <-> 
( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
20 3orrot 951 . . . . . . . . 9  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C )  \/  ( ph  /\  x  =  A ) ) )
21 df-3or 946 . . . . . . . . 9  |-  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C )  \/  ( ph  /\  x  =  A ) )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) )
2220, 21bitri 183 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) )
2319, 22syl6bbr 197 . . . . . . 7  |-  ( ph  ->  ( x  =  A  <-> 
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
2423eubidv 1983 . . . . . 6  |-  ( ph  ->  ( E! x  x  =  A  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
254, 24mpbii 147 . . . . 5  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
26 eueq3dc.3 . . . . . . 7  |-  C  e. 
_V
2726eueq1 2827 . . . . . 6  |-  E! x  x  =  C
28 ibar 297 . . . . . . . . 9  |-  ( ps 
->  ( x  =  C  <-> 
( ps  /\  x  =  C ) ) )
298adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  =  A )  ->  -.  ps )
30 pm2.46 711 . . . . . . . . . . . . 13  |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
3130adantr 272 . . . . . . . . . . . 12  |-  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  ->  -.  ps )
3229, 31jaoi 688 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  ->  -.  ps )
3332con2i 599 . . . . . . . . . 10  |-  ( ps 
->  -.  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )
) )
34 biorf 716 . . . . . . . . . 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 187 . . . . . . . 8  |-  ( ps 
->  ( x  =  C  <-> 
( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )
)  \/  ( ps 
/\  x  =  C ) ) ) )
37 df-3or 946 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) )
3836, 37syl6bbr 197 . . . . . . 7  |-  ( ps 
->  ( x  =  C  <-> 
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
3938eubidv 1983 . . . . . 6  |-  ( ps 
->  ( E! x  x  =  C  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
4027, 39mpbii 147 . . . . 5  |-  ( ps 
->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
4125, 40jaoi 688 . . . 4  |-  ( (
ph  \/  ps )  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
42 eueq3dc.2 . . . . . 6  |-  B  e. 
_V
4342eueq1 2827 . . . . 5  |-  E! x  x  =  B
44 ibar 297 . . . . . . . 8  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( -.  ( ph  \/  ps )  /\  x  =  B )
) )
45 simpl 108 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  ph )
46 simpl 108 . . . . . . . . . . 11  |-  ( ( ps  /\  x  =  C )  ->  ps )
4745, 46orim12i 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  ->  ( ph  \/  ps ) )
4847con3i 604 . . . . . . . . 9  |-  ( -.  ( ph  \/  ps )  ->  -.  ( ( ph  /\  x  =  A )  \/  ( ps 
/\  x  =  C ) ) )
49 biorf 716 . . . . . . . . 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 187 . . . . . . 7  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( ( (
ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
52 3orcomb 954 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( ps 
/\  x  =  C )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
53 df-3or 946 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
5452, 53bitri 183 . . . . . . 7  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
5551, 54syl6bbr 197 . . . . . 6  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
5655eubidv 1983 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  ( E! x  x  =  B  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
5743, 56mpbii 147 . . . 4  |-  ( -.  ( ph  \/  ps )  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
5841, 57jaoi 688 . . 3  |-  ( ( ( ph  \/  ps )  \/  -.  ( ph  \/  ps ) )  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
592, 58sylbi 120 . 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 103    <-> wb 104    \/ wo 680  DECID wdc 802    \/ w3o 944    = wceq 1314    e. wcel 1463   E!weu 1975   _Vcvv 2658
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  moeq3dc  2831
  Copyright terms: Public domain W3C validator