Theorem subeq0d

Description: If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subeq0d.3	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 0 )
Assertion	subeq0d	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subeq0d.3	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 0 )
4		subeq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
5	1 2 4	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
6	3 5	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )