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Metamath Proof Explorer

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 )