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Theorem tgcgrneq 25599
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 25597 . . 3 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
1211necon3bid 2987 . 2 (𝜑 → (𝐴𝐵𝐶𝐷))
131, 12mpbid 222 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgc 25568  df-trkg 25573
This theorem is referenced by:  hlcgrex  25732  midexlem  25808  footex  25834  mideulem2  25847  opphllem3  25862  trgcopy  25917  iscgra1  25923  cgrane1  25925  cgrane2  25926  cgrcgra  25934  cgrg3col4  25955  tgsas2  25958  tgsas3  25959  tgasa1  25960  tgsss1  25962
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