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Theorem axtgcgrid 23986
 Description: Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p
axtrkg.d
axtrkg.i Itv
axtrkg.g TarskiG
axtgcgrid.1
axtgcgrid.2
axtgcgrid.3
axtgcgrid.4
Assertion
Ref Expression
axtgcgrid

Proof of Theorem axtgcgrid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 23976 . . . . 5 TarskiG TarskiGC TarskiGB TarskiGCB Itv LineG
2 inss1 3714 . . . . . 6 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC TarskiGB
3 inss1 3714 . . . . . 6 TarskiGC TarskiGB TarskiGC
42, 3sstri 3508 . . . . 5 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC
51, 4eqsstri 3529 . . . 4 TarskiG TarskiGC
6 axtrkg.g . . . 4 TarskiG
75, 6sseldi 3497 . . 3 TarskiGC
8 axtrkg.p . . . . . 6
9 axtrkg.d . . . . . 6
10 axtrkg.i . . . . . 6 Itv
118, 9, 10istrkgc 23977 . . . . 5 TarskiGC
1211simprbi 464 . . . 4 TarskiGC
1312simprd 463 . . 3 TarskiGC
147, 13syl 16 . 2
15 axtgcgrid.4 . 2
16 axtgcgrid.1 . . 3
17 axtgcgrid.2 . . 3
18 axtgcgrid.3 . . 3
19 oveq1 6303 . . . . . 6
2019eqeq1d 2459 . . . . 5
21 eqeq1 2461 . . . . 5
2220, 21imbi12d 320 . . . 4
23 oveq2 6304 . . . . . 6
2423eqeq1d 2459 . . . . 5
25 eqeq2 2472 . . . . 5
2624, 25imbi12d 320 . . . 4
27 id 22 . . . . . . 7
2827, 27oveq12d 6314 . . . . . 6
2928eqeq2d 2471 . . . . 5
3029imbi1d 317 . . . 4
3122, 26, 30rspc3v 3222 . . 3
3216, 17, 18, 31syl3anc 1228 . 2
3314, 15, 32mp2d 45 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3o 972   wceq 1395   wcel 1819  cab 2442  wral 2807  crab 2811  cvv 3109  wsbc 3327   cdif 3468   cin 3470  csn 4032  cfv 5594  (class class class)co 6296   cmpt2 6298  cbs 14644  cds 14721  TarskiGcstrkg 23951  TarskiGCcstrkgc 23952  TarskiGBcstrkgb 23953  TarskiGCBcstrkgcb 23954  Itvcitv 23958  LineGclng 23959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-trkgc 23970  df-trkg 23976 This theorem is referenced by:  tgcgreqb  23998  tgcgrtriv  24001  tgsegconeq  24003  tgbtwntriv2  24004  tgbtwndiff  24023  tgifscgr  24026  tgbtwnxfr  24044  lnid  24083  tgbtwnconn1lem2  24086  tgbtwnconn1lem3  24087  legtri3  24103  legeq  24106  legbtwn  24107  mirreu3  24161  colmid  24191  krippenlem  24193  lmiisolem  24287  hypcgrlem1  24290  hypcgrlem2  24291  f1otrg  24301
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