This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )