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Theorem eueq2dc 2828
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 803 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 notnot 601 . . . . 5  |-  ( ph  ->  -.  -.  ph )
3 eueq2dc.1 . . . . . . 7  |-  A  e. 
_V
43eueq1 2827 . . . . . 6  |-  E! x  x  =  A
5 euanv 2032 . . . . . . 7  |-  ( E! x ( ph  /\  x  =  A )  <->  (
ph  /\  E! x  x  =  A )
)
65biimpri 132 . . . . . 6  |-  ( (
ph  /\  E! x  x  =  A )  ->  E! x ( ph  /\  x  =  A ) )
74, 6mpan2 419 . . . . 5  |-  ( ph  ->  E! x ( ph  /\  x  =  A ) )
8 euorv 2002 . . . . 5  |-  ( ( -.  -.  ph  /\  E! x ( ph  /\  x  =  A )
)  ->  E! x
( -.  ph  \/  ( ph  /\  x  =  A ) ) )
92, 7, 8syl2anc 406 . . . 4  |-  ( ph  ->  E! x ( -. 
ph  \/  ( ph  /\  x  =  A ) ) )
10 orcom 700 . . . . . 6  |-  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  -.  ph ) )
112bianfd 915 . . . . . . 7  |-  ( ph  ->  ( -.  ph  <->  ( -.  ph 
/\  x  =  B ) ) )
1211orbi2d 762 . . . . . 6  |-  ( ph  ->  ( ( ( ph  /\  x  =  A )  \/  -.  ph )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
1310, 12syl5bb 191 . . . . 5  |-  ( ph  ->  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -. 
ph  /\  x  =  B ) ) ) )
1413eubidv 1983 . . . 4  |-  ( ph  ->  ( E! x ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
159, 14mpbid 146 . . 3  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
16 eueq2dc.2 . . . . . . 7  |-  B  e. 
_V
1716eueq1 2827 . . . . . 6  |-  E! x  x  =  B
18 euanv 2032 . . . . . . 7  |-  ( E! x ( -.  ph  /\  x  =  B )  <-> 
( -.  ph  /\  E! x  x  =  B ) )
1918biimpri 132 . . . . . 6  |-  ( ( -.  ph  /\  E! x  x  =  B )  ->  E! x ( -. 
ph  /\  x  =  B ) )
2017, 19mpan2 419 . . . . 5  |-  ( -. 
ph  ->  E! x ( -.  ph  /\  x  =  B ) )
21 euorv 2002 . . . . 5  |-  ( ( -.  ph  /\  E! x
( -.  ph  /\  x  =  B )
)  ->  E! x
( ph  \/  ( -.  ph  /\  x  =  B ) ) )
2220, 21mpdan 415 . . . 4  |-  ( -. 
ph  ->  E! x (
ph  \/  ( -.  ph 
/\  x  =  B ) ) )
23 id 19 . . . . . . 7  |-  ( -. 
ph  ->  -.  ph )
2423bianfd 915 . . . . . 6  |-  ( -. 
ph  ->  ( ph  <->  ( ph  /\  x  =  A ) ) )
2524orbi1d 763 . . . . 5  |-  ( -. 
ph  ->  ( ( ph  \/  ( -.  ph  /\  x  =  B )
)  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
2625eubidv 1983 . . . 4  |-  ( -. 
ph  ->  ( E! x
( ph  \/  ( -.  ph  /\  x  =  B ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
2722, 26mpbid 146 . . 3  |-  ( -. 
ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
2815, 27jaoi 688 . 2  |-  ( (
ph  \/  -.  ph )  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
291, 28sylbi 120 1  |-  (DECID  ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = 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-tru 1317  df-fal 1320  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: (None)
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