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Theorem eueq2 3110
Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
Hypotheses
Ref Expression
eueq2.1  |-  A  e. 
_V
eueq2.2  |-  B  e. 
_V
Assertion
Ref Expression
eueq2  |-  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
)
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem eueq2
StepHypRef Expression
1 notnot1 117 . . . 4  |-  ( ph  ->  -.  -.  ph )
2 eueq2.1 . . . . . 6  |-  A  e. 
_V
32eueq1 3109 . . . . 5  |-  E! x  x  =  A
4 euanv 2344 . . . . . 6  |-  ( E! x ( ph  /\  x  =  A )  <->  (
ph  /\  E! x  x  =  A )
)
54biimpri 199 . . . . 5  |-  ( (
ph  /\  E! x  x  =  A )  ->  E! x ( ph  /\  x  =  A ) )
63, 5mpan2 654 . . . 4  |-  ( ph  ->  E! x ( ph  /\  x  =  A ) )
7 euorv 2311 . . . 4  |-  ( ( -.  -.  ph  /\  E! x ( ph  /\  x  =  A )
)  ->  E! x
( -.  ph  \/  ( ph  /\  x  =  A ) ) )
81, 6, 7syl2anc 644 . . 3  |-  ( ph  ->  E! x ( -. 
ph  \/  ( ph  /\  x  =  A ) ) )
9 orcom 378 . . . . 5  |-  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  -.  ph ) )
101bianfd 894 . . . . . 6  |-  ( ph  ->  ( -.  ph  <->  ( -.  ph 
/\  x  =  B ) ) )
1110orbi2d 684 . . . . 5  |-  ( ph  ->  ( ( ( ph  /\  x  =  A )  \/  -.  ph )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
129, 11syl5bb 250 . . . 4  |-  ( ph  ->  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -. 
ph  /\  x  =  B ) ) ) )
1312eubidv 2291 . . 3  |-  ( ph  ->  ( E! x ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
148, 13mpbid 203 . 2  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
15 eueq2.2 . . . . . 6  |-  B  e. 
_V
1615eueq1 3109 . . . . 5  |-  E! x  x  =  B
17 euanv 2344 . . . . . 6  |-  ( E! x ( -.  ph  /\  x  =  B )  <-> 
( -.  ph  /\  E! x  x  =  B ) )
1817biimpri 199 . . . . 5  |-  ( ( -.  ph  /\  E! x  x  =  B )  ->  E! x ( -. 
ph  /\  x  =  B ) )
1916, 18mpan2 654 . . . 4  |-  ( -. 
ph  ->  E! x ( -.  ph  /\  x  =  B ) )
20 euorv 2311 . . . 4  |-  ( ( -.  ph  /\  E! x
( -.  ph  /\  x  =  B )
)  ->  E! x
( ph  \/  ( -.  ph  /\  x  =  B ) ) )
2119, 20mpdan 651 . . 3  |-  ( -. 
ph  ->  E! x (
ph  \/  ( -.  ph 
/\  x  =  B ) ) )
22 id 21 . . . . . 6  |-  ( -. 
ph  ->  -.  ph )
2322bianfd 894 . . . . 5  |-  ( -. 
ph  ->  ( ph  <->  ( ph  /\  x  =  A ) ) )
2423orbi1d 685 . . . 4  |-  ( -. 
ph  ->  ( ( ph  \/  ( -.  ph  /\  x  =  B )
)  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
2524eubidv 2291 . . 3  |-  ( -. 
ph  ->  ( E! x
( ph  \/  ( -.  ph  /\  x  =  B ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
2621, 25mpbid 203 . 2  |-  ( -. 
ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
2714, 26pm2.61i 159 1  |-  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2283   _Vcvv 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
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