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Theorem isig2a2 25478
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 25477 . . 3  |-  ( ph  ->  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
) ) )
8 neeq1 2454 . . . . 5  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
9 eleq1 2343 . . . . . . 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 2724 . . . . 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 2455 . . . . 5  |-  ( y  =  B  ->  ( A  =/=  y  <->  A  =/=  B ) )
14 eleq1 2343 . . . . . . 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 2724 . . . . 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 2891 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E!wreu 2545   ` cfv 5255  PPointscpoints 25468  PLinescplines 25470  Igcig 25472
This theorem is referenced by:  lineval42  25492  lineval12  25493  lineval22  25494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ig2 25473
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