Theorem subadd2i

Description: Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
	subadd.3	⊢ 𝐶 ∈ ℂ
Assertion	subadd2i	⊢ ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		subadd.3	⊢ 𝐶 ∈ ℂ
4		subadd2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐶 + 𝐵 ) = 𝐴 )