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Theorem tgcgrneq 25178
Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgcgrcomlr.a (𝜑𝐴𝑃)
tgcgrcomlr.b (𝜑𝐵𝑃)
tgcgrcomlr.c (𝜑𝐶𝑃)
tgcgrcomlr.d (𝜑𝐷𝑃)
tgcgrcomlr.6 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
tgcgrneq.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
tgcgrneq (𝜑𝐶𝐷)

Proof of Theorem tgcgrneq
StepHypRef Expression
1 tgcgrneq.1 . 2 (𝜑𝐴𝐵)
2 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
3 tkgeom.d . . . 4 = (dist‘𝐺)
4 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
6 tgcgrcomlr.a . . . 4 (𝜑𝐴𝑃)
7 tgcgrcomlr.b . . . 4 (𝜑𝐵𝑃)
8 tgcgrcomlr.c . . . 4 (𝜑𝐶𝑃)
9 tgcgrcomlr.d . . . 4 (𝜑𝐷𝑃)
10 tgcgrcomlr.6 . . . 4 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
112, 3, 4, 5, 6, 7, 8, 9, 10tgcgreqb 25176 . . 3 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
1211necon3bid 2826 . 2 (𝜑 → (𝐴𝐵𝐶𝐷))
131, 12mpbid 221 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wne 2780  cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkg 25152
This theorem is referenced by:  hlcgrex  25311  midexlem  25387  footex  25413  mideulem2  25426  opphllem3  25441  trgcopy  25496  iscgra1  25502  cgrane1  25504  cgrane2  25505  cgrcgra  25513  cgrg3col4  25534  tgsas2  25537  tgsas3  25538  tgasa1  25539  tgsss1  25541
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