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Theorem tgcgrcoml 26192
Description: Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgcgrcomr.a (𝜑𝐴𝑃)
tgcgrcomr.b (𝜑𝐵𝑃)
tgcgrcomr.c (𝜑𝐶𝑃)
tgcgrcomr.d (𝜑𝐷𝑃)
tgcgrcomr.6 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
Assertion
Ref Expression
tgcgrcoml (𝜑 → (𝐵 𝐴) = (𝐶 𝐷))

Proof of Theorem tgcgrcoml
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 . . 3 (𝜑𝐺 ∈ TarskiG)
5 tgcgrcomr.a . . 3 (𝜑𝐴𝑃)
6 tgcgrcomr.b . . 3 (𝜑𝐵𝑃)
71, 2, 3, 4, 5, 6axtgcgrrflx 26175 . 2 (𝜑 → (𝐴 𝐵) = (𝐵 𝐴))
8 tgcgrcomr.6 . 2 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
97, 8eqtr3d 2855 1 (𝜑 → (𝐵 𝐴) = (𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgc 26161  df-trkg 26166
This theorem is referenced by:  hlcgrex  26329  dfcgra2  26543
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