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Theorem tgcgrcomr 25572
Description: Congruence commutes on the RHS. 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
tgcgrcomr (𝜑 → (𝐴 𝐵) = (𝐷 𝐶))

Proof of Theorem tgcgrcomr
StepHypRef Expression
1 tgcgrcomr.6 . 2 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
2 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
3 tkgeom.d . . 3 = (dist‘𝐺)
4 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . 3 (𝜑𝐺 ∈ TarskiG)
6 tgcgrcomr.c . . 3 (𝜑𝐶𝑃)
7 tgcgrcomr.d . . 3 (𝜑𝐷𝑃)
82, 3, 4, 5, 6, 7axtgcgrrflx 25560 . 2 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
91, 8eqtrd 2794 1 (𝜑 → (𝐴 𝐵) = (𝐷 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6813  Basecbs 16059  distcds 16152  TarskiGcstrkg 25528  Itvcitv 25534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-trkgc 25546  df-trkg 25551
This theorem is referenced by:  tgbtwnconn1lem1  25666  dfcgra2  25920
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