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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