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Metamath Proof Explorer

Theorem subne0ad

Description: If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d . Contrapositive of subeq0bd . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subne0ad.3	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ≠ 0 )
Assertion	subne0ad	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subne0ad.3	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ≠ 0 )
4	1 2	subeq0ad	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
5	4	necon3bid	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) ≠ 0 ↔ 𝐴 ≠ 𝐵 ) )
6	3 5	mpbid	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )