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Theorem tgcgrneq 28506
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 28504 . . 3 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
1211necon3bid 2983 . 2 (𝜑 → (𝐴𝐵𝐶𝐷))
131, 12mpbid 232 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkgc 28471  df-trkg 28476
This theorem is referenced by:  hlcgrex  28639  midexlem  28715  footexALT  28741  footexlem1  28742  footexlem2  28743  mideulem2  28757  opphllem3  28772  trgcopy  28827  iscgra1  28833  cgrane1  28835  cgrane2  28836  cgrcgra  28844  flatcgra  28847  cgrg3col4  28876  tgsas2  28879  tgsas3  28880  tgasa1  28881
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