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Theorem tgcgrcoml 25174
 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 25161 . 2 (𝜑 → (𝐴 𝐵) = (𝐵 𝐴))
8 tgcgrcomr.6 . 2 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
97, 8eqtr3d 2646 1 (𝜑 → (𝐵 𝐴) = (𝐶 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkg 25152 This theorem is referenced by:  hlcgrex  25311  dfcgra2  25521
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