Theorem tgcgrcomimp 28450

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrcomimp.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrcomimp.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrcomimp.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrcomimp.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)

Assertion

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

Proof of Theorem tgcgrcomimp

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgcgrcomimp.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
6		tgcgrcomimp.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
7	1, 2, 3, 4, 5, 6	axtgcgrrflx 28435	. . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
8	7	eqeq2d 2742	. 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐶)))
9	8	biimpd 229	1 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  distcds 17165  TarskiGcstrkg 28400  Itvcitv 28406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-trkgc 28421  df-trkg 28426

This theorem is referenced by: (None)