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Theorem tgcgreqb 26387
 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))
Assertion
Ref Expression
tgcgreqb (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Proof of Theorem tgcgreqb
StepHypRef Expression
1 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
2 tkgeom.d . . 3 = (dist‘𝐺)
3 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54adantr 484 . . 3 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
6 tgcgrcomlr.c . . . 4 (𝜑𝐶𝑃)
76adantr 484 . . 3 ((𝜑𝐴 = 𝐵) → 𝐶𝑃)
8 tgcgrcomlr.d . . . 4 (𝜑𝐷𝑃)
98adantr 484 . . 3 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
10 tgcgrcomlr.b . . . 4 (𝜑𝐵𝑃)
1110adantr 484 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
12 tgcgrcomlr.6 . . . . 5 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
1312adantr 484 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐶 𝐷))
14 simpr 488 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
1514oveq1d 7171 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐵 𝐵))
1613, 15eqtr3d 2795 . . 3 ((𝜑𝐴 = 𝐵) → (𝐶 𝐷) = (𝐵 𝐵))
171, 2, 3, 5, 7, 9, 11, 16axtgcgrid 26369 . 2 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐷)
184adantr 484 . . 3 ((𝜑𝐶 = 𝐷) → 𝐺 ∈ TarskiG)
19 tgcgrcomlr.a . . . 4 (𝜑𝐴𝑃)
2019adantr 484 . . 3 ((𝜑𝐶 = 𝐷) → 𝐴𝑃)
2110adantr 484 . . 3 ((𝜑𝐶 = 𝐷) → 𝐵𝑃)
228adantr 484 . . 3 ((𝜑𝐶 = 𝐷) → 𝐷𝑃)
2312adantr 484 . . . 4 ((𝜑𝐶 = 𝐷) → (𝐴 𝐵) = (𝐶 𝐷))
24 simpr 488 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
2524oveq1d 7171 . . . 4 ((𝜑𝐶 = 𝐷) → (𝐶 𝐷) = (𝐷 𝐷))
2623, 25eqtrd 2793 . . 3 ((𝜑𝐶 = 𝐷) → (𝐴 𝐵) = (𝐷 𝐷))
271, 2, 3, 18, 20, 21, 22, 26axtgcgrid 26369 . 2 ((𝜑𝐶 = 𝐷) → 𝐴 = 𝐵)
2817, 27impbida 800 1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  distcds 16645  TarskiGcstrkg 26336  Itvcitv 26342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5180 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-trkgc 26354  df-trkg 26359 This theorem is referenced by:  tgcgreq  26388  tgcgrneq  26389
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