Theorem negsubdi2i

Description: Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
Assertion	negsubdi2i	⊢ - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3	1 2	negsubdii	⊢ - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 )
4	1	negcli	⊢ - 𝐴 ∈ ℂ
5	2 1	negsubi	⊢ ( 𝐵 + - 𝐴 ) = ( 𝐵 − 𝐴 )
6	2 4 5	addcomli	⊢ ( - 𝐴 + 𝐵 ) = ( 𝐵 − 𝐴 )
7	3 6	eqtri	⊢ - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 )