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Theorem tgcgrneq 25285
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 25283 . . 3 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
1211necon3bid 2834 . 2 (𝜑 → (𝐴𝐵𝐶𝐷))
131, 12mpbid 222 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wne 2790  cfv 5849  (class class class)co 6607  Basecbs 15784  distcds 15874  TarskiGcstrkg 25236  Itvcitv 25242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-iota 5812  df-fv 5857  df-ov 6610  df-trkgc 25254  df-trkg 25259
This theorem is referenced by:  hlcgrex  25418  midexlem  25494  footex  25520  mideulem2  25533  opphllem3  25548  trgcopy  25603  iscgra1  25609  cgrane1  25611  cgrane2  25612  cgrcgra  25620  cgrg3col4  25641  tgsas2  25644  tgsas3  25645  tgasa1  25646  tgsss1  25648
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