Theorem subaddrii

Description: Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
	subadd.3	⊢ 𝐶 ∈ ℂ
	subaddri.4	⊢ ( 𝐵 + 𝐶 ) = 𝐴
Assertion	subaddrii	⊢ ( 𝐴 − 𝐵 ) = 𝐶

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		subadd.3	⊢ 𝐶 ∈ ℂ
4		subaddri.4	⊢ ( 𝐵 + 𝐶 ) = 𝐴
5	1 2 3	subaddi	⊢ ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 )
6	4 5	mpbir	⊢ ( 𝐴 − 𝐵 ) = 𝐶