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Theorem tgcgrcomimp 25062
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 25051 . . 3 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
87eqeq2d 2524 . 2 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) ↔ (𝐴 𝐵) = (𝐷 𝐶)))
98biimpd 217 1 (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) → (𝐴 𝐵) = (𝐷 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  cfv 5689  (class class class)co 6425  Basecbs 15577  distcds 15659  TarskiGcstrkg 25019  Itvcitv 25025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-nul 4616
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697  df-ov 6428  df-trkgc 25037  df-trkg 25042
This theorem is referenced by: (None)
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