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Theorem eueq2dc 2766
Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
Hypotheses
Ref Expression
eueq2dc.1  |-  A  e. 
_V
eueq2dc.2  |-  B  e. 
_V
Assertion
Ref Expression
eueq2dc  |-  (DECID  ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem eueq2dc
StepHypRef Expression
1 df-dc 777 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 notnot 592 . . . . 5  |-  ( ph  ->  -.  -.  ph )
3 eueq2dc.1 . . . . . . 7  |-  A  e. 
_V
43eueq1 2765 . . . . . 6  |-  E! x  x  =  A
5 euanv 1999 . . . . . . 7  |-  ( E! x ( ph  /\  x  =  A )  <->  (
ph  /\  E! x  x  =  A )
)
65biimpri 131 . . . . . 6  |-  ( (
ph  /\  E! x  x  =  A )  ->  E! x ( ph  /\  x  =  A ) )
74, 6mpan2 416 . . . . 5  |-  ( ph  ->  E! x ( ph  /\  x  =  A ) )
8 euorv 1969 . . . . 5  |-  ( ( -.  -.  ph  /\  E! x ( ph  /\  x  =  A )
)  ->  E! x
( -.  ph  \/  ( ph  /\  x  =  A ) ) )
92, 7, 8syl2anc 403 . . . 4  |-  ( ph  ->  E! x ( -. 
ph  \/  ( ph  /\  x  =  A ) ) )
10 orcom 680 . . . . . 6  |-  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  -.  ph ) )
112bianfd 890 . . . . . . 7  |-  ( ph  ->  ( -.  ph  <->  ( -.  ph 
/\  x  =  B ) ) )
1211orbi2d 737 . . . . . 6  |-  ( ph  ->  ( ( ( ph  /\  x  =  A )  \/  -.  ph )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
1310, 12syl5bb 190 . . . . 5  |-  ( ph  ->  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -. 
ph  /\  x  =  B ) ) ) )
1413eubidv 1950 . . . 4  |-  ( ph  ->  ( E! x ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
159, 14mpbid 145 . . 3  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
16 eueq2dc.2 . . . . . . 7  |-  B  e. 
_V
1716eueq1 2765 . . . . . 6  |-  E! x  x  =  B
18 euanv 1999 . . . . . . 7  |-  ( E! x ( -.  ph  /\  x  =  B )  <-> 
( -.  ph  /\  E! x  x  =  B ) )
1918biimpri 131 . . . . . 6  |-  ( ( -.  ph  /\  E! x  x  =  B )  ->  E! x ( -. 
ph  /\  x  =  B ) )
2017, 19mpan2 416 . . . . 5  |-  ( -. 
ph  ->  E! x ( -.  ph  /\  x  =  B ) )
21 euorv 1969 . . . . 5  |-  ( ( -.  ph  /\  E! x
( -.  ph  /\  x  =  B )
)  ->  E! x
( ph  \/  ( -.  ph  /\  x  =  B ) ) )
2220, 21mpdan 412 . . . 4  |-  ( -. 
ph  ->  E! x (
ph  \/  ( -.  ph 
/\  x  =  B ) ) )
23 id 19 . . . . . . 7  |-  ( -. 
ph  ->  -.  ph )
2423bianfd 890 . . . . . 6  |-  ( -. 
ph  ->  ( ph  <->  ( ph  /\  x  =  A ) ) )
2524orbi1d 738 . . . . 5  |-  ( -. 
ph  ->  ( ( ph  \/  ( -.  ph  /\  x  =  B )
)  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
2625eubidv 1950 . . . 4  |-  ( -. 
ph  ->  ( E! x
( ph  \/  ( -.  ph  /\  x  =  B ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
2722, 26mpbid 145 . . 3  |-  ( -. 
ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
2815, 27jaoi 669 . 2  |-  ( (
ph  \/  -.  ph )  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
291, 28sylbi 119 1  |-  (DECID  ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   E!weu 1942   _Vcvv 2602
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by: (None)
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