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Theorem lineval42 26183
 Description: Any line to which and are incident is the line . (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
Hypotheses
Ref Expression
lineval2.1 PPoints
lineval2.2 PLines
lineval2.3
lineval2.4 Ig
lineval42.5
lineval42.6
lineval42.7
lineval42.8
lineval42.9
lineval42.a
Assertion
Ref Expression
lineval42

Proof of Theorem lineval42
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lineval2.1 . . 3 PPoints
2 lineval2.2 . . 3 PLines
3 lineval2.3 . . 3
4 lineval2.4 . . 3 Ig
5 lineval42.5 . . 3
6 lineval42.6 . . 3
7 lineval42.7 . . 3
81, 2, 3, 4, 5, 6, 7lineval222 26182 . 2
9 lineval42.8 . . 3
10 lineval42.9 . . 3
11 lineval42.a . . . 4
121, 2, 4, 5, 6, 7isig2a2 26169 . . . 4
13 eleq2 2357 . . . . . 6
14 eleq2 2357 . . . . . 6
1513, 14anbi12d 691 . . . . 5
1615riota2 6343 . . . 4
1711, 12, 16syl2anc 642 . . 3
189, 10, 17mpbi2and 887 . 2
198, 18eqtr2d 2329 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1632   wcel 1696   wne 2459  wreu 2558  cfv 5271  (class class class)co 5874  crio 6313  PPointscpoints 26159  PLinescplines 26161  Igcig 26163  cline 26179 This theorem is referenced by:  lineval5a  26191  lineval6a  26192  isconcl5a  26204  isconcl5ab  26205  isibg1a6  26228 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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 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-mo 2161  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-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-ig2 26164  df-li 26180
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