Theorem negcon1ad

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

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	negcon1ad.2	⊢ ( 𝜑 → - 𝐴 = 𝐵 )
Assertion	negcon1ad	⊢ ( 𝜑 → - 𝐵 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		negcon1ad.2	⊢ ( 𝜑 → - 𝐴 = 𝐵 )
3	1	negcld	⊢ ( 𝜑 → - 𝐴 ∈ ℂ )
4	2 3	eqeltrrd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
5	1 4	negcon1d	⊢ ( 𝜑 → ( - 𝐴 = 𝐵 ↔ - 𝐵 = 𝐴 ) )
6	2 5	mpbid	⊢ ( 𝜑 → - 𝐵 = 𝐴 )