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Theorem lineval42 26080
Description: Any line to which  A and  B are incident is the line  ( A M B ). (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
Hypotheses
Ref Expression
lineval2.1  |-  P  =  (PPoints `  G )
lineval2.2  |-  L  =  (PLines `  G )
lineval2.3  |-  M  =  ( line `  G
)
lineval2.4  |-  ( ph  ->  G  e. Ig )
lineval42.5  |-  ( ph  ->  A  e.  P )
lineval42.6  |-  ( ph  ->  B  e.  P )
lineval42.7  |-  ( ph  ->  A  =/=  B )
lineval42.8  |-  ( ph  ->  A  e.  N )
lineval42.9  |-  ( ph  ->  B  e.  N )
lineval42.a  |-  ( ph  ->  N  e.  L )
Assertion
Ref Expression
lineval42  |-  ( ph  ->  N  =  ( A M B ) )

Proof of Theorem lineval42
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 lineval2.1 . . 3  |-  P  =  (PPoints `  G )
2 lineval2.2 . . 3  |-  L  =  (PLines `  G )
3 lineval2.3 . . 3  |-  M  =  ( line `  G
)
4 lineval2.4 . . 3  |-  ( ph  ->  G  e. Ig )
5 lineval42.5 . . 3  |-  ( ph  ->  A  e.  P )
6 lineval42.6 . . 3  |-  ( ph  ->  B  e.  P )
7 lineval42.7 . . 3  |-  ( ph  ->  A  =/=  B )
81, 2, 3, 4, 5, 6, 7lineval222 26079 . 2  |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
9 lineval42.8 . . 3  |-  ( ph  ->  A  e.  N )
10 lineval42.9 . . 3  |-  ( ph  ->  B  e.  N )
11 lineval42.a . . . 4  |-  ( ph  ->  N  e.  L )
121, 2, 4, 5, 6, 7isig2a2 26066 . . . 4  |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )
13 eleq2 2344 . . . . . 6  |-  ( l  =  N  ->  ( A  e.  l  <->  A  e.  N ) )
14 eleq2 2344 . . . . . 6  |-  ( l  =  N  ->  ( B  e.  l  <->  B  e.  N ) )
1513, 14anbi12d 691 . . . . 5  |-  ( l  =  N  ->  (
( A  e.  l  /\  B  e.  l )  <->  ( A  e.  N  /\  B  e.  N ) ) )
1615riota2 6327 . . . 4  |-  ( ( N  e.  L  /\  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )  ->  (
( A  e.  N  /\  B  e.  N
)  <->  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  =  N ) )
1711, 12, 16syl2anc 642 . . 3  |-  ( ph  ->  ( ( A  e.  N  /\  B  e.  N )  <->  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  =  N ) )
189, 10, 17mpbi2and 887 . 2  |-  ( ph  ->  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )  =  N )
198, 18eqtr2d 2316 1  |-  ( ph  ->  N  =  ( A M B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E!wreu 2545   ` cfv 5255  (class class class)co 5858   iota_crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  lineval5a  26088  lineval6a  26089  isconcl5a  26101  isconcl5ab  26102  isibg1a6  26125
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-mo 2148  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-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ig2 26061  df-li 26077
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