Theorem npncan2

Description: Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013)

	Ref	Expression
Assertion	npncan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + ( 𝐵 − 𝐴 ) ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		npncan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐴 − 𝐴 ) )
2	1	3anidm13	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐴 − 𝐴 ) )
3		subid	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 𝐴 ) = 0 )
4	3	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 𝐴 ) = 0 )
5	2 4	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + ( 𝐵 − 𝐴 ) ) = 0 )