Theorem negcon1d

Description: Contraposition law for unary minus. Deduction form of negcon1 . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	negcon1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	negcon1d	⊢ ( 𝜑 → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		negcon1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		negcon1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )