Theorem tgcgrcoml 25565

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 25552	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴))
8		tgcgrcomr.6	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
9	7, 8	eqtr3d 2788	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷))

Syntax hints:   → wi 4   = wceq 1624   ∈ wcel 2131  ‘cfv 6041  (class class class)co 6805  Basecbs 16051  distcds 16144  TarskiGcstrkg 25520  Itvcitv 25526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-nul 4933

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-iota 6004  df-fv 6049  df-ov 6808  df-trkgc 25538  df-trkg 25543

This theorem is referenced by:  hlcgrex  25702  dfcgra2  25912