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Theorem tgcgreqb 23736
 Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Hypotheses
Ref Expression
tkgeom.p
tkgeom.d
tkgeom.i Itv
tkgeom.g TarskiG
tgcgrcomlr.a
tgcgrcomlr.b
tgcgrcomlr.c
tgcgrcomlr.d
tgcgrcomlr.6
Assertion
Ref Expression
tgcgreqb

Proof of Theorem tgcgreqb
StepHypRef Expression
1 tkgeom.p . . 3
2 tkgeom.d . . 3
3 tkgeom.i . . 3 Itv
4 tkgeom.g . . . 4 TarskiG
54adantr 465 . . 3 TarskiG
6 tgcgrcomlr.c . . . 4
8 tgcgrcomlr.d . . . 4
10 tgcgrcomlr.b . . . 4
12 tgcgrcomlr.6 . . . . 5
1312adantr 465 . . . 4
14 simpr 461 . . . . 5
1514oveq1d 6310 . . . 4
1613, 15eqtr3d 2510 . . 3
171, 2, 3, 5, 7, 9, 11, 16axtgcgrid 23724 . 2
184adantr 465 . . 3 TarskiG
19 tgcgrcomlr.a . . . 4
2312adantr 465 . . . 4
24 simpr 461 . . . . 5
2524oveq1d 6310 . . . 4
2623, 25eqtrd 2508 . . 3
271, 2, 3, 18, 20, 21, 22, 26axtgcgrid 23724 . 2
2817, 27impbida 830 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  cfv 5594  (class class class)co 6295  cbs 14506  cds 14580  TarskiGcstrkg 23689  Itvcitv 23696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-trkgc 23708  df-trkg 23714 This theorem is referenced by:  tgcgreq  23737  tgcgrneq  23738
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