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Theorem tgcgrcomimp 28499
Description: Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. (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)
tgcgrcomimp.a (𝜑𝐴𝑃)
tgcgrcomimp.b (𝜑𝐵𝑃)
tgcgrcomimp.c (𝜑𝐶𝑃)
tgcgrcomimp.d (𝜑𝐷𝑃)
Assertion
Ref Expression
tgcgrcomimp (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) → (𝐴 𝐵) = (𝐷 𝐶)))

Proof of Theorem tgcgrcomimp
StepHypRef Expression
1 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . 4 = (dist‘𝐺)
3 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
5 tgcgrcomimp.c . . . 4 (𝜑𝐶𝑃)
6 tgcgrcomimp.d . . . 4 (𝜑𝐷𝑃)
71, 2, 3, 4, 5, 6axtgcgrrflx 28484 . . 3 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
87eqeq2d 2745 . 2 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐷 𝐶)))
98biimpd 229 1 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) → (𝐴 𝐵) = (𝐷 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cfv 6562  (class class class)co 7430  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  Itvcitv 28455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-trkgc 28470  df-trkg 28475
This theorem is referenced by: (None)
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