Theorem subeq0

Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999)

	Ref	Expression
Assertion	subeq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		subid	⊢ ( 𝐵 ∈ ℂ → ( 𝐵 − 𝐵 ) = 0 )
2	1	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 − 𝐵 ) = 0 )
3	2	eqeq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐵 ) ↔ ( 𝐴 − 𝐵 ) = 0 ) )
4		subcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	4	3anidm23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐵 ) ↔ 𝐴 = 𝐵 ) )
6	3 5	bitr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )