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Theorem tgcgrcomimp 26423
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 26408 . . 3 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
87eqeq2d 2749 . 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 1542  wcel 2114  cfv 6339  (class class class)co 7170  Basecbs 16586  distcds 16677  TarskiGcstrkg 26376  Itvcitv 26382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7173  df-trkgc 26394  df-trkg 26399
This theorem is referenced by: (None)
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