Theorem tgcgrcomimp 28746

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 28731	. . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
8	7	eqeq2d 2774	. 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐶)))
9	8	biimpd 232	1 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶)))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  distcds 17314  TarskiGcstrkg 28696  Itvcitv 28702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-trkgc 28717  df-trkg 28722

This theorem is referenced by: (None)