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Theorem tgcgrcomimp 28152
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 28137 . . 3 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
87eqeq2d 2735 . 2 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐷 𝐶)))
98biimpd 228 1 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) → (𝐴 𝐵) = (𝐷 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6533  (class class class)co 7401  Basecbs 17140  distcds 17202  TarskiGcstrkg 28102  Itvcitv 28108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-trkgc 28123  df-trkg 28128
This theorem is referenced by: (None)
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