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Metamath Proof Explorer

Theorem neg11ad

Description: The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 . Generalization of neg11d . (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses negidd.1 ⊢ φ → A ∈ ℂ
neg11ad.2 ⊢ φ → B ∈ ℂ
Assertion neg11ad ⊢ φ → − A = − B ↔ A = B

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 neg11ad.2 ⊢ φ → B ∈ ℂ
3 neg11 ⊢ A ∈ ℂ ∧ B ∈ ℂ → − A = − B ↔ A = B
4 1 2 3 syl2anc ⊢ φ → − A = − B ↔ A = B