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Theorem eueq2 3076
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 116 . . . 4  |-  ( ph  ->  -.  -.  ph )
2 eueq2.1 . . . . . 6  |-  A  e. 
_V
32eueq1 3075 . . . . 5  |-  E! x  x  =  A
4 euanv 2323 . . . . . 6  |-  ( E! x ( ph  /\  x  =  A )  <->  (
ph  /\  E! x  x  =  A )
)
54biimpri 198 . . . . 5  |-  ( (
ph  /\  E! x  x  =  A )  ->  E! x ( ph  /\  x  =  A ) )
63, 5mpan2 653 . . . 4  |-  ( ph  ->  E! x ( ph  /\  x  =  A ) )
7 euorv 2290 . . . 4  |-  ( ( -.  -.  ph  /\  E! x ( ph  /\  x  =  A )
)  ->  E! x
( -.  ph  \/  ( ph  /\  x  =  A ) ) )
81, 6, 7syl2anc 643 . . 3  |-  ( ph  ->  E! x ( -. 
ph  \/  ( ph  /\  x  =  A ) ) )
9 orcom 377 . . . . 5  |-  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  -.  ph ) )
101bianfd 893 . . . . . 6  |-  ( ph  ->  ( -.  ph  <->  ( -.  ph 
/\  x  =  B ) ) )
1110orbi2d 683 . . . . 5  |-  ( ph  ->  ( ( ( ph  /\  x  =  A )  \/  -.  ph )  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
129, 11syl5bb 249 . . . 4  |-  ( ph  ->  ( ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  ( ( ph  /\  x  =  A )  \/  ( -. 
ph  /\  x  =  B ) ) ) )
1312eubidv 2270 . . 3  |-  ( ph  ->  ( E! x ( -.  ph  \/  ( ph  /\  x  =  A ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
148, 13mpbid 202 . 2  |-  ( ph  ->  E! x ( (
ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
15 eueq2.2 . . . . . 6  |-  B  e. 
_V
1615eueq1 3075 . . . . 5  |-  E! x  x  =  B
17 euanv 2323 . . . . . 6  |-  ( E! x ( -.  ph  /\  x  =  B )  <-> 
( -.  ph  /\  E! x  x  =  B ) )
1817biimpri 198 . . . . 5  |-  ( ( -.  ph  /\  E! x  x  =  B )  ->  E! x ( -. 
ph  /\  x  =  B ) )
1916, 18mpan2 653 . . . 4  |-  ( -. 
ph  ->  E! x ( -.  ph  /\  x  =  B ) )
20 euorv 2290 . . . 4  |-  ( ( -.  ph  /\  E! x
( -.  ph  /\  x  =  B )
)  ->  E! x
( ph  \/  ( -.  ph  /\  x  =  B ) ) )
2119, 20mpdan 650 . . 3  |-  ( -. 
ph  ->  E! x (
ph  \/  ( -.  ph 
/\  x  =  B ) ) )
22 id 20 . . . . . 6  |-  ( -. 
ph  ->  -.  ph )
2322bianfd 893 . . . . 5  |-  ( -. 
ph  ->  ( ph  <->  ( ph  /\  x  =  A ) ) )
2423orbi1d 684 . . . 4  |-  ( -. 
ph  ->  ( ( ph  \/  ( -.  ph  /\  x  =  B )
)  <->  ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) ) )
2524eubidv 2270 . . 3  |-  ( -. 
ph  ->  ( E! x
( ph  \/  ( -.  ph  /\  x  =  B ) )  <->  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
) ) )
2621, 25mpbid 202 . 2  |-  ( -. 
ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
2714, 26pm2.61i 158 1  |-  E! x
( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   E!weu 2262   _Vcvv 2924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-v 2926
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