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Theorem tgcgreq 28490
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))
tgcgreq.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
tgcgreq (𝜑𝐶 = 𝐷)

Proof of Theorem tgcgreq
StepHypRef Expression
1 tgcgreq.1 . 2 (𝜑𝐴 = 𝐵)
2 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
3 tkgeom.d . . 3 = (dist‘𝐺)
4 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . 3 (𝜑𝐺 ∈ TarskiG)
6 tgcgrcomlr.a . . 3 (𝜑𝐴𝑃)
7 tgcgrcomlr.b . . 3 (𝜑𝐵𝑃)
8 tgcgrcomlr.c . . 3 (𝜑𝐶𝑃)
9 tgcgrcomlr.d . . 3 (𝜑𝐷𝑃)
10 tgcgrcomlr.6 . . 3 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
112, 3, 4, 5, 6, 7, 8, 9, 10tgcgreqb 28489 . 2 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
121, 11mpbid 232 1 (𝜑𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgc 28456  df-trkg 28461
This theorem is referenced by:  tgcgrextend  28493  tgidinside  28579  tgbtwnconn1lem3  28582  krippenlem  28698  ragcgr  28715  lmiisolem  28804  cgrg3col4  28861
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