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Metamath Proof Explorer

Theorem neg11ad

Description: The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 . Generalization of neg11d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	neg11ad.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	neg11ad	⊢ ( 𝜑 → ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		neg11ad.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		neg11	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( - 𝐴 = - 𝐵 ↔ 𝐴 = 𝐵 ) )