Theorem tgcgrcomimp 28939

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 28924	. . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
8	7	eqeq2d 2772	. 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 1570   ∈ wcel 2145  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  Itvcitv 28895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-trkgc 28910  df-trkg 28915

This theorem is used by: (None)