Theorem tgcgrcoml 27128

Description: Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrcomr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrcomr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrcomr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrcomr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgrcomr.6	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))

Assertion

Ref	Expression
tgcgrcoml	⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷))

Proof of Theorem tgcgrcoml

Step	Hyp	Ref	Expression
1		tkgeom.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. . 3 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgcgrcomr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		tgcgrcomr.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	1, 2, 3, 4, 5, 6	axtgcgrrflx 27111	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴))
8		tgcgrcomr.6	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
9	7, 8	eqtr3d 2779	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6483  (class class class)co 7341  Basecbs 17009  distcds 17068  TarskiGcstrkg 27076  Itvcitv 27082

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5254

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rab 3405  df-v 3444  df-sbc 3731  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-iota 6435  df-fv 6491  df-ov 7344  df-trkgc 27097  df-trkg 27102

This theorem is referenced by:  hlcgrex  27265  dfcgra2  27479