Theorem negsubdii

Description: Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999)

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

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3	2	negcli	⊢ - 𝐵 ∈ ℂ
4	1 3	negdii	⊢ - ( 𝐴 + - 𝐵 ) = ( - 𝐴 + - - 𝐵 )
5	1 2	negsubi	⊢ ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 )
6	5	negeqi	⊢ - ( 𝐴 + - 𝐵 ) = - ( 𝐴 − 𝐵 )
7	2	negnegi	⊢ - - 𝐵 = 𝐵
8	7	oveq2i	⊢ ( - 𝐴 + - - 𝐵 ) = ( - 𝐴 + 𝐵 )
9	4 6 8	3eqtr3i	⊢ - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 )