Theorem subeq0i

Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
Assertion	subeq0i	⊢ ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		subeq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
4	1 2 3	mp2an	⊢ ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 )