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Theorem eueq3 2953
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
eueq3.1  |-  A  e. 
_V
eueq3.2  |-  B  e. 
_V
eueq3.3  |-  C  e. 
_V
eueq3.4  |-  -.  ( ph  /\  ps )
Assertion
Ref Expression
eueq3  |-  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 eueq3
StepHypRef Expression
1 eueq3.1 . . . 4  |-  A  e. 
_V
21eueq1 2951 . . 3  |-  E! x  x  =  A
3 ibar 490 . . . . . 6  |-  ( ph  ->  ( x  =  A  <-> 
( ph  /\  x  =  A ) ) )
4 pm2.45 386 . . . . . . . . . 10  |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
5 eueq3.4 . . . . . . . . . . . 12  |-  -.  ( ph  /\  ps )
65imnani 412 . . . . . . . . . . 11  |-  ( ph  ->  -.  ps )
76con2i 112 . . . . . . . . . 10  |-  ( ps 
->  -.  ph )
84, 7jaoi 368 . . . . . . . . 9  |-  ( ( -.  ( ph  \/  ps )  \/  ps )  ->  -.  ph )
98con2i 112 . . . . . . . 8  |-  ( ph  ->  -.  ( -.  ( ph  \/  ps )  \/ 
ps ) )
104con2i 112 . . . . . . . . . 10  |-  ( ph  ->  -.  -.  ( ph  \/  ps ) )
1110bianfd 892 . . . . . . . . 9  |-  ( ph  ->  ( -.  ( ph  \/  ps )  <->  ( -.  ( ph  \/  ps )  /\  x  =  B
) ) )
126bianfd 892 . . . . . . . . 9  |-  ( ph  ->  ( ps  <->  ( ps  /\  x  =  C ) ) )
1311, 12orbi12d 690 . . . . . . . 8  |-  ( ph  ->  ( ( -.  ( ph  \/  ps )  \/ 
ps )  <->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) ) )
149, 13mtbid 291 . . . . . . 7  |-  ( ph  ->  -.  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
15 biorf 394 . . . . . . 7  |-  ( -.  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  -> 
( ( ph  /\  x  =  A )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
1614, 15syl 15 . . . . . 6  |-  ( ph  ->  ( ( ph  /\  x  =  A )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
173, 16bitrd 244 . . . . 5  |-  ( ph  ->  ( x  =  A  <-> 
( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) ) )
18 3orrot 940 . . . . . 6  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C )  \/  ( ph  /\  x  =  A ) ) )
19 df-3or 935 . . . . . 6  |-  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C )  \/  ( ph  /\  x  =  A ) )  <->  ( ( ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) )
2018, 19bitri 240 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )  \/  ( ph  /\  x  =  A ) ) )
2117, 20syl6bbr 254 . . . 4  |-  ( ph  ->  ( x  =  A  <-> 
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
2221eubidv 2164 . . 3  |-  ( ph  ->  ( E! x  x  =  A  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
232, 22mpbii 202 . 2  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
24 eueq3.3 . . . 4  |-  C  e. 
_V
2524eueq1 2951 . . 3  |-  E! x  x  =  C
26 ibar 490 . . . . . 6  |-  ( ps 
->  ( x  =  C  <-> 
( ps  /\  x  =  C ) ) )
276adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  -.  ps )
28 pm2.46 387 . . . . . . . . . 10  |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
2928adantr 451 . . . . . . . . 9  |-  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  ->  -.  ps )
3027, 29jaoi 368 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  ->  -.  ps )
3130con2i 112 . . . . . . 7  |-  ( ps 
->  -.  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )
) )
32 biorf 394 . . . . . . 7  |-  ( -.  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  -> 
( ( ps  /\  x  =  C )  <->  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) ) )
3331, 32syl 15 . . . . . 6  |-  ( ps 
->  ( ( ps  /\  x  =  C )  <->  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) ) )
3426, 33bitrd 244 . . . . 5  |-  ( ps 
->  ( x  =  C  <-> 
( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )
)  \/  ( ps 
/\  x  =  C ) ) ) )
35 df-3or 935 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  \/  ( ps  /\  x  =  C ) ) )
3634, 35syl6bbr 254 . . . 4  |-  ( ps 
->  ( x  =  C  <-> 
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
3736eubidv 2164 . . 3  |-  ( ps 
->  ( E! x  x  =  C  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
3825, 37mpbii 202 . 2  |-  ( ps 
->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
39 eueq3.2 . . . 4  |-  B  e. 
_V
4039eueq1 2951 . . 3  |-  E! x  x  =  B
41 ibar 490 . . . . . 6  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( -.  ( ph  \/  ps )  /\  x  =  B )
) )
42 simpl 443 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  ph )
43 simpl 443 . . . . . . . . 9  |-  ( ( ps  /\  x  =  C )  ->  ps )
4442, 43orim12i 502 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  ->  ( ph  \/  ps ) )
4544con3i 127 . . . . . . 7  |-  ( -.  ( ph  \/  ps )  ->  -.  ( ( ph  /\  x  =  A )  \/  ( ps 
/\  x  =  C ) ) )
46 biorf 394 . . . . . . 7  |-  ( -.  ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  -> 
( ( -.  ( ph  \/  ps )  /\  x  =  B )  <->  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
4745, 46syl 15 . . . . . 6  |-  ( -.  ( ph  \/  ps )  ->  ( ( -.  ( ph  \/  ps )  /\  x  =  B )  <->  ( ( (
ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
4841, 47bitrd 244 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( ( (
ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) ) )
49 3orcomb 944 . . . . . 6  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( ( ph  /\  x  =  A )  \/  ( ps 
/\  x  =  C )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
50 df-3or 935 . . . . . 6  |-  ( ( ( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
5149, 50bitri 240 . . . . 5  |-  ( ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) )  <->  ( (
( ph  /\  x  =  A )  \/  ( ps  /\  x  =  C ) )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B ) ) )
5248, 51syl6bbr 254 . . . 4  |-  ( -.  ( ph  \/  ps )  ->  ( x  =  B  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
5352eubidv 2164 . . 3  |-  ( -.  ( ph  \/  ps )  ->  ( E! x  x  =  B  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) ) ) )
5440, 53mpbii 202 . 2  |-  ( -.  ( ph  \/  ps )  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps 
/\  x  =  C ) ) )
5523, 38, 54ecase3 907 1  |-  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   E!weu 2156   _Vcvv 2801
This theorem is referenced by:  moeq3  2955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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