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Theorem axtgcgrrflx 23723
 Description: Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p
axtrkg.d
axtrkg.i Itv
axtrkg.g TarskiG
axtgcgrrflx.1
axtgcgrrflx.2
Assertion
Ref Expression
axtgcgrrflx

Proof of Theorem axtgcgrrflx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 23714 . . . . 5 TarskiG TarskiGC TarskiGB TarskiGCB Itv LineG
2 inss1 3723 . . . . . 6 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC TarskiGB
3 inss1 3723 . . . . . 6 TarskiGC TarskiGB TarskiGC
42, 3sstri 3518 . . . . 5 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC
51, 4eqsstri 3539 . . . 4 TarskiG TarskiGC
6 axtrkg.g . . . 4 TarskiG
75, 6sseldi 3507 . . 3 TarskiGC
8 axtrkg.p . . . . . 6
9 axtrkg.d . . . . . 6
10 axtrkg.i . . . . . 6 Itv
118, 9, 10istrkgc 23715 . . . . 5 TarskiGC
1211simprbi 464 . . . 4 TarskiGC
1312simpld 459 . . 3 TarskiGC
147, 13syl 16 . 2
15 axtgcgrrflx.1 . . 3
16 axtgcgrrflx.2 . . 3
17 oveq1 6302 . . . . 5
18 oveq2 6303 . . . . 5
1917, 18eqeq12d 2489 . . . 4
20 oveq2 6303 . . . . 5
21 oveq1 6302 . . . . 5
2220, 21eqeq12d 2489 . . . 4
2319, 22rspc2v 3228 . . 3
2415, 16, 23syl2anc 661 . 2
2514, 24mpd 15 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3o 972   wceq 1379   wcel 1767  cab 2452  wral 2817  crab 2821  cvv 3118  wsbc 3336   cdif 3478   cin 3480  csn 4033  cfv 5594  (class class class)co 6295   cmpt2 6297  cbs 14506  cds 14580  TarskiGcstrkg 23689  TarskiGCcstrkgc 23690  TarskiGBcstrkgb 23691  TarskiGCBcstrkgcb 23692  Itvcitv 23696  LineGclng 23697 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:  tgcgrcomlr  23735  tgbtwnconn1lem1  23822  tgbtwnconn1lem2  23823  tgbtwnconn1lem3  23824  miriso  23908  symquadlem  23921  midexlem  23924  footex  23950  colperpexlem1  23956  opphllem  23961  f1otrg  24006
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