Theorem tgcgrcomimp

Description: Congruence commutes on the RHS. Theorem 2.5 of Schwabhauser p. 27. (Contributed by David A. Wheeler, 29-Jun-2020)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgcgrcomimp.a	|- ( ph -> A e. P )
	tgcgrcomimp.b	|- ( ph -> B e. P )
	tgcgrcomimp.c	|- ( ph -> C e. P )
	tgcgrcomimp.d	|- ( ph -> D e. P )
Assertion	tgcgrcomimp	|- ( ph -> ( ( A .- B ) = ( C .- D ) -> ( A .- B ) = ( D .- C ) ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	|- P = ( Base ` G )
2		tkgeom.d	|- .- = ( dist ` G )
3		tkgeom.i	|- I = ( Itv ` G )
4		tkgeom.g	|- ( ph -> G e. TarskiG )
5		tgcgrcomimp.a	|- ( ph -> A e. P )
6		tgcgrcomimp.b	|- ( ph -> B e. P )
7		tgcgrcomimp.c	|- ( ph -> C e. P )
8		tgcgrcomimp.d	|- ( ph -> D e. P )
9	1 2 3 4 7 8	axtgcgrrflx	|- ( ph -> ( C .- D ) = ( D .- C ) )
10	9	eqeq2d	|- ( ph -> ( ( A .- B ) = ( C .- D ) <-> ( A .- B ) = ( D .- C ) ) )
11	10	biimpd	|- ( ph -> ( ( A .- B ) = ( C .- D ) -> ( A .- B ) = ( D .- C ) ) )