Theorem subeq0ad

Description: The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 . Generalization of subeq0d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	subeq0ad	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )

Proof

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