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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgcgrcomimp.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgcgrcomimp.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgcgrcomimp.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgcgrcomimp.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
Assertion	tgcgrcomimp	⊢ ( 𝜑 → ( ( 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgcgrcomimp.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgcgrcomimp.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgcgrcomimp.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgcgrcomimp.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9	1 2 3 4 7 8	axtgcgrrflx	⊢ ( 𝜑 → ( 𝐶 − 𝐷 ) = ( 𝐷 − 𝐶 ) )
10	9	eqeq2d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) ↔ ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐶 ) ) )
11	10	biimpd	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐶 ) ) )