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Theorem tgcgrcoml 25565
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 25552 . 2 (𝜑 → (𝐴 𝐵) = (𝐵 𝐴))
8 tgcgrcomr.6 . 2 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
97, 8eqtr3d 2788 1 (𝜑 → (𝐵 𝐴) = (𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131  cfv 6041  (class class class)co 6805  Basecbs 16051  distcds 16144  TarskiGcstrkg 25520  Itvcitv 25526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-nul 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-iota 6004  df-fv 6049  df-ov 6808  df-trkgc 25538  df-trkg 25543
This theorem is referenced by:  hlcgrex  25702  dfcgra2  25912
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