Theorem tgcgrcomimp 28948

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 28933	. . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
8	7	eqeq2d 2771	. 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐶)))
9	8	biimpd 232	1 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  distcds 17446  TarskiGcstrkg 28898  Itvcitv 28904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427  df-trkgc 28919  df-trkg 28924

This theorem is used by: (None)