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Theorem istrkgc 23976
 Description: Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
istrkg.p
istrkg.d
istrkg.i Itv
Assertion
Ref Expression
istrkgc TarskiGC
Distinct variable groups:   ,,,   ,,,   , ,,
Allowed substitution hints:   (,,)

Proof of Theorem istrkgc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istrkg.p . . 3
2 istrkg.d . . 3
3 simpl 457 . . . . . 6
43eqcomd 2465 . . . . 5
54adantr 465 . . . . . 6
6 simpllr 760 . . . . . . . . 9
76eqcomd 2465 . . . . . . . 8
87oveqd 6313 . . . . . . 7
97oveqd 6313 . . . . . . 7
108, 9eqeq12d 2479 . . . . . 6
115, 10raleqbidva 3070 . . . . 5
124, 11raleqbidva 3070 . . . 4
135adantr 465 . . . . . . 7
147oveqdr 6320 . . . . . . . . 9
157oveqdr 6320 . . . . . . . . 9
1614, 15eqeq12d 2479 . . . . . . . 8
1716imbi1d 317 . . . . . . 7
1813, 17raleqbidva 3070 . . . . . 6
195, 18raleqbidva 3070 . . . . 5
204, 19raleqbidva 3070 . . . 4
2112, 20anbi12d 710 . . 3
221, 2, 21sbcie2s 14688 . 2
23 df-trkgc 23969 . 2 TarskiGC
2422, 23elab4g 3250 1 TarskiGC
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807  cvv 3109  wsbc 3327  cfv 5594  (class class class)co 6296  cbs 14643  cds 14720  TarskiGCcstrkgc 23951  Itvcitv 23957 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 23969 This theorem is referenced by:  axtgcgrrflx  23984  axtgcgrid  23985  f1otrg  24300  xmstrkgc  24315  eengtrkg  24414
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