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Theorem tgcgreq 25793
 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 25792 . 2 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
121, 11mpbid 224 1 (𝜑𝐶 = 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ‘cfv 6122  (class class class)co 6904  Basecbs 16221  distcds 16313  TarskiGcstrkg 25741  Itvcitv 25747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-nul 5012 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-iota 6085  df-fv 6130  df-ov 6907  df-trkgc 25759  df-trkg 25764 This theorem is referenced by:  tgcgrextend  25796  tgidinside  25882  tgbtwnconn1lem3  25885  krippenlem  26001  ragcgr  26018  lmiisolem  26104  cgrg3col4  26151
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