Theorem tgcgrcoml 25274

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 25261	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴))
8		tgcgrcomr.6	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
9	7, 8	eqtr3d 2657	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷))

Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-trkgc 25247  df-trkg 25252

This theorem is referenced by:  hlcgrex  25411  dfcgra2  25621