Theorem negsubdi2

Description: Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999)

	Ref	Expression
Assertion	negsubdi2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		negsubdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( - 𝐴 + 𝐵 ) )
2		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
3		addcom	⊢ ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 + 𝐵 ) = ( 𝐵 + - 𝐴 ) )
4	2 3	sylan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 + 𝐵 ) = ( 𝐵 + - 𝐴 ) )
5		negsub	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 + - 𝐴 ) = ( 𝐵 − 𝐴 ) )
6	5	ancoms	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 + - 𝐴 ) = ( 𝐵 − 𝐴 ) )
7	1 4 6	3eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 ) )