Theorem subeqxfrd

Description: Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020)

	Ref	Expression
Hypotheses	subeqxfrd.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	subeqxfrd.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subeqxfrd.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	subeqxfrd.d	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
	subeqxfrd.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
Assertion	subeqxfrd	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐵 − 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		subeqxfrd.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		subeqxfrd.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subeqxfrd.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		subeqxfrd.d	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5		subeqxfrd.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
6	5	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) + ( 𝐵 − 𝐶 ) ) = ( ( 𝐶 − 𝐷 ) + ( 𝐵 − 𝐶 ) ) )
7	1 2 3	npncand	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) + ( 𝐵 − 𝐶 ) ) = ( 𝐴 − 𝐶 ) )
8	3 4 2	npncan3d	⊢ ( 𝜑 → ( ( 𝐶 − 𝐷 ) + ( 𝐵 − 𝐶 ) ) = ( 𝐵 − 𝐷 ) )
9	6 7 8	3eqtr3d	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐵 − 𝐷 ) )