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Theorem isig2a2 26169
Description: There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Hypotheses
Ref Expression
isig.1  |-  P  =  (PPoints `  I )
isig.2  |-  L  =  (PLines `  I )
isig2a2.1  |-  ( ph  ->  I  e. Ig )
isig2a2.2  |-  ( ph  ->  A  e.  P )
isig2a2.3  |-  ( ph  ->  B  e.  P )
isig2a2.4  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
isig2a2  |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )
Distinct variable groups:    A, l    B, l    L, l    P, l
Allowed substitution hints:    ph( l)    I(
l)

Proof of Theorem isig2a2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isig2a2.4 . 2  |-  ( ph  ->  A  =/=  B )
2 isig2a2.2 . . 3  |-  ( ph  ->  A  e.  P )
3 isig2a2.3 . . 3  |-  ( ph  ->  B  e.  P )
4 isig.1 . . . 4  |-  P  =  (PPoints `  I )
5 isig.2 . . . 4  |-  L  =  (PLines `  I )
6 isig2a2.1 . . . 4  |-  ( ph  ->  I  e. Ig )
74, 5, 6isig22 26168 . . 3  |-  ( ph  ->  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
) ) )
8 neeq1 2467 . . . . 5  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
9 eleq1 2356 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  l  <->  A  e.  l ) )
109anbi1d 685 . . . . . 6  |-  ( x  =  A  ->  (
( x  e.  l  /\  y  e.  l )  <->  ( A  e.  l  /\  y  e.  l ) ) )
1110reubidv 2737 . . . . 5  |-  ( x  =  A  ->  ( E! l  e.  L  ( x  e.  l  /\  y  e.  l
)  <->  E! l  e.  L  ( A  e.  l  /\  y  e.  l
) ) )
128, 11imbi12d 311 . . . 4  |-  ( x  =  A  ->  (
( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
) )  <->  ( A  =/=  y  ->  E! l  e.  L  ( A  e.  l  /\  y  e.  l ) ) ) )
13 neeq2 2468 . . . . 5  |-  ( y  =  B  ->  ( A  =/=  y  <->  A  =/=  B ) )
14 eleq1 2356 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  l  <->  B  e.  l ) )
1514anbi2d 684 . . . . . 6  |-  ( y  =  B  ->  (
( A  e.  l  /\  y  e.  l )  <->  ( A  e.  l  /\  B  e.  l ) ) )
1615reubidv 2737 . . . . 5  |-  ( y  =  B  ->  ( E! l  e.  L  ( A  e.  l  /\  y  e.  l
)  <->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) ) )
1713, 16imbi12d 311 . . . 4  |-  ( y  =  B  ->  (
( A  =/=  y  ->  E! l  e.  L  ( A  e.  l  /\  y  e.  l
) )  <->  ( A  =/=  B  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l ) ) ) )
1812, 17rspc2va 2904 . . 3  |-  ( ( ( A  e.  P  /\  B  e.  P
)  /\  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l ) ) )  ->  ( A  =/= 
B  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l ) ) )
192, 3, 7, 18syl21anc 1181 . 2  |-  ( ph  ->  ( A  =/=  B  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) ) )
201, 19mpd 14 1  |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E!wreu 2558   ` cfv 5271  PPointscpoints 26159  PLinescplines 26161  Igcig 26163
This theorem is referenced by:  lineval42  26183  lineval12  26184  lineval22  26185
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  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ig2 26164
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