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Theorem eupath2lem2 23917
Description: Lemma for eupath2 23919. (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 2297 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =  U )  ->  B  =  B )
21olcd 382 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  \/  B  =  B ) )
32biantrud 493 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =/=  B  <->  ( A  =/=  B  /\  ( B  =  A  \/  B  =  B
) ) ) )
4 eupath2lem2.1 . . . . . 6  |-  B  e. 
_V
5 eupath2lem1 23916 . . . . . 6  |-  ( B  e.  _V  ->  ( B  e.  if ( A  =  B ,  (/)
,  { A ,  B } )  <->  ( A  =/=  B  /\  ( B  =  A  \/  B  =  B ) ) ) )
64, 5ax-mp 8 . . . . 5  |-  ( B  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  ( A  =/=  B  /\  ( B  =  A  \/  B  =  B ) ) )
73, 6syl6bbr 254 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =/=  B  <->  B  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
8 simpr 447 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  B  =  U )
98eleq1d 2362 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
107, 9bitrd 244 . . 3  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =/=  B  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
1110necon1bbid 2513 . 2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
)  <->  A  =  B
) )
12 simpl 443 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =  U )  ->  B  =/=  C )
13 neeq1 2467 . . . . . . 7  |-  ( B  =  A  ->  ( B  =/=  C  <->  A  =/=  C ) )
1412, 13syl5ibcom 211 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  ->  A  =/=  C
) )
1514pm4.71rd 616 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  <-> 
( A  =/=  C  /\  B  =  A
) ) )
16 eqcom 2298 . . . . 5  |-  ( A  =  B  <->  B  =  A )
17 ancom 437 . . . . 5  |-  ( ( B  =  A  /\  A  =/=  C )  <->  ( A  =/=  C  /\  B  =  A ) )
1815, 16, 173bitr4g 279 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =  B  <-> 
( B  =  A  /\  A  =/=  C
) ) )
1912neneqd 2475 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =  U )  ->  -.  B  =  C )
20 biorf 394 . . . . . . 7  |-  ( -.  B  =  C  -> 
( B  =  A  <-> 
( B  =  C  \/  B  =  A ) ) )
2119, 20syl 15 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  <-> 
( B  =  C  \/  B  =  A ) ) )
22 orcom 376 . . . . . 6  |-  ( ( B  =  C  \/  B  =  A )  <->  ( B  =  A  \/  B  =  C )
)
2321, 22syl6bb 252 . . . . 5  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  =  A  <-> 
( B  =  A  \/  B  =  C ) ) )
2423anbi1d 685 . . . 4  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( ( B  =  A  /\  A  =/= 
C )  <->  ( ( B  =  A  \/  B  =  C )  /\  A  =/=  C
) ) )
2518, 24bitrd 244 . . 3  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =  B  <-> 
( ( B  =  A  \/  B  =  C )  /\  A  =/=  C ) ) )
26 ancom 437 . . 3  |-  ( ( A  =/=  C  /\  ( B  =  A  \/  B  =  C
) )  <->  ( ( B  =  A  \/  B  =  C )  /\  A  =/=  C
) )
2725, 26syl6bbr 254 . 2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( A  =  B  <-> 
( A  =/=  C  /\  ( B  =  A  \/  B  =  C ) ) ) )
28 eupath2lem1 23916 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  if ( A  =  C ,  (/)
,  { A ,  C } )  <->  ( A  =/=  C  /\  ( B  =  A  \/  B  =  C ) ) ) )
294, 28ax-mp 8 . . 3  |-  ( B  e.  if ( A  =  C ,  (/) ,  { A ,  C } )  <->  ( A  =/=  C  /\  ( B  =  A  \/  B  =  C ) ) )
308eleq1d 2362 . . 3  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( B  e.  if ( A  =  C ,  (/) ,  { A ,  C } )  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
3129, 30syl5bbr 250 . 2  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( ( A  =/= 
C  /\  ( B  =  A  \/  B  =  C ) )  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
3211, 27, 313bitrd 270 1  |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
)  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   ifcif 3578   {cpr 3654
This theorem is referenced by:  eupath2lem3  23918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660
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