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Theorem tgcgrcomimp 26249
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 26234 . . 3 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
87eqeq2d 2832 . 2 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐷 𝐶)))
98biimpd 232 1 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) → (𝐴 𝐵) = (𝐷 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16461  distcds 16552  TarskiGcstrkg 26202  Itvcitv 26208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-trkgc 26220  df-trkg 26225
This theorem is referenced by: (None)
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