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	$$⊢\phi \to A\in ℂ$$
	negcon1ad.2	$$⊢\phi \to -A=B$$
Assertion	negcon1ad	$$⊢\phi \to -B=A$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		negcon1ad.2	$$⊢\phi \to -A=B$$
3	1	negcld	$$⊢\phi \to -A\in ℂ$$
4	2 3	eqeltrrd	$$⊢\phi \to B\in ℂ$$
5	1 4	negcon1d	$$⊢\phi \to \left(-A=B\leftrightarrow -B=A\right)$$
6	2 5	mpbid	$$⊢\phi \to -B=A$$