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Theorem eupath2lem2 23597
Description: Lemma for eupath2 23599. (Contributed by Mario Carneiro, 8-Apr-2015.)
Hypothesis
Ref Expression
eupath2lem2.1  |-  B  e. 
_V
Assertion
Ref Expression
eupath2lem2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
)  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )

Proof of Theorem eupath2lem2
StepHypRef Expression
1 eqidd 2442 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =  U )  ->  B  =  B )
21olcd 393 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  \/  B  =  B ) )
32biantrud 507 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =/=  B  <->  ( A  =/=  B  /\  ( B  =  A  \/  B  =  B
) ) ) )
4 eupath2lem2.1 . . . . . 6  |-  B  e. 
_V
5 eupath2lem1 23596 . . . . . 6  |-  ( B  e.  _V  ->  ( B  e.  if ( A  =  B ,  (/)
,  { A ,  B } )  <->  ( A  =/=  B  /\  ( B  =  A  \/  B  =  B ) ) ) )
64, 5ax-mp 5 . . . . 5  |-  ( B  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  ( A  =/=  B  /\  ( B  =  A  \/  B  =  B ) ) )
73, 6syl6bbr 263 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =/=  B  <->  B  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
8 simpr 461 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  B  =  U )
98eleq1d 2507 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
107, 9bitrd 253 . . 3  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =/=  B  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
1110necon1bbid 2663 . 2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
)  <->  A  =  B
) )
12 simpl 457 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =  U )  ->  B  =/=  C )
13 neeq1 2614 . . . . . . 7  |-  ( B  =  A  ->  ( B  =/=  C  <->  A  =/=  C ) )
1412, 13syl5ibcom 220 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  ->  A  =/=  C
) )
1514pm4.71rd 635 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  <-> 
( A  =/=  C  /\  B  =  A
) ) )
16 eqcom 2443 . . . . 5  |-  ( A  =  B  <->  B  =  A )
17 ancom 450 . . . . 5  |-  ( ( B  =  A  /\  A  =/=  C )  <->  ( A  =/=  C  /\  B  =  A ) )
1815, 16, 173bitr4g 288 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =  B  <-> 
( B  =  A  /\  A  =/=  C
) ) )
1912neneqd 2622 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =  U )  ->  -.  B  =  C )
20 biorf 405 . . . . . . 7  |-  ( -.  B  =  C  -> 
( B  =  A  <-> 
( B  =  C  \/  B  =  A ) ) )
2119, 20syl 16 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  <-> 
( B  =  C  \/  B  =  A ) ) )
22 orcom 387 . . . . . 6  |-  ( ( B  =  C  \/  B  =  A )  <->  ( B  =  A  \/  B  =  C )
)
2321, 22syl6bb 261 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  <-> 
( B  =  A  \/  B  =  C ) ) )
2423anbi1d 704 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( ( B  =  A  /\  A  =/= 
C )  <->  ( ( B  =  A  \/  B  =  C )  /\  A  =/=  C
) ) )
2518, 24bitrd 253 . . 3  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =  B  <-> 
( ( B  =  A  \/  B  =  C )  /\  A  =/=  C ) ) )
26 ancom 450 . . 3  |-  ( ( A  =/=  C  /\  ( B  =  A  \/  B  =  C
) )  <->  ( ( B  =  A  \/  B  =  C )  /\  A  =/=  C
) )
2725, 26syl6bbr 263 . 2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =  B  <-> 
( A  =/=  C  /\  ( B  =  A  \/  B  =  C ) ) ) )
28 eupath2lem1 23596 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  if ( A  =  C ,  (/)
,  { A ,  C } )  <->  ( A  =/=  C  /\  ( B  =  A  \/  B  =  C ) ) ) )
294, 28ax-mp 5 . . 3  |-  ( B  e.  if ( A  =  C ,  (/) ,  { A ,  C } )  <->  ( A  =/=  C  /\  ( B  =  A  \/  B  =  C ) ) )
308eleq1d 2507 . . 3  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  e.  if ( A  =  C ,  (/) ,  { A ,  C } )  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
3129, 30syl5bbr 259 . 2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( ( A  =/= 
C  /\  ( B  =  A  \/  B  =  C ) )  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
3211, 27, 313bitrd 279 1  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
)  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970   (/)c0 3635   ifcif 3789   {cpr 3877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-v 2972  df-dif 3329  df-un 3331  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878
This theorem is referenced by:  eupath2lem3  23598
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