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Theorem axtgcgrid 23616
 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 23606 . . . . 5 TarskiG TarskiGC TarskiGB TarskiGCB Itv LineG
2 inss1 3718 . . . . . 6 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC TarskiGB
3 inss1 3718 . . . . . 6 TarskiGC TarskiGB TarskiGC
42, 3sstri 3513 . . . . 5 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC
51, 4eqsstri 3534 . . . 4 TarskiG TarskiGC
6 axtrkg.g . . . 4 TarskiG
75, 6sseldi 3502 . . 3 TarskiGC
8 axtrkg.p . . . . . 6
9 axtrkg.d . . . . . 6
10 axtrkg.i . . . . . 6 Itv
118, 9, 10istrkgc 23607 . . . . 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 6291 . . . . . 6
2019eqeq1d 2469 . . . . 5
21 eqeq1 2471 . . . . 5
2220, 21imbi12d 320 . . . 4
23 oveq2 6292 . . . . . 6
2423eqeq1d 2469 . . . . 5
25 eqeq2 2482 . . . . 5
2624, 25imbi12d 320 . . . 4
27 id 22 . . . . . . 7
2827, 27oveq12d 6302 . . . . . 6
2928eqeq2d 2481 . . . . 5
3029imbi1d 317 . . . 4
3122, 26, 30rspc3v 3226 . . 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 1379   wcel 1767  cab 2452  wral 2814  crab 2818  cvv 3113  wsbc 3331   cdif 3473   cin 3475  csn 4027  cfv 5588  (class class class)co 6284   cmpt2 6286  cbs 14490  cds 14564  TarskiGcstrkg 23581  TarskiGCcstrkgc 23582  TarskiGBcstrkgb 23583  TarskiGCBcstrkgcb 23584  Itvcitv 23588  LineGclng 23589 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 4576 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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287  df-trkgc 23600  df-trkg 23606 This theorem is referenced by:  tgcgreqb  23628  tgcgrtriv  23631  tgcgrextend  23632  tgsegconeq  23633  tgbtwntriv2  23634  tgbtwndiff  23653  tgifscgr  23656  tgbtwnxfr  23674  lnid  23712  tgidinside  23713  tgbtwnconn1lem2  23715  tgbtwnconn1lem3  23716  legtri3  23732  legeq  23735  legbtwn  23736  mirreu3  23776  colmid  23801  krippenlem  23803  lmimid  23864  lmiisolem  23866  hypcgrlem1  23869  hypcgrlem2  23870  f1otrg  23878
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