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Metamath Proof Explorer

Theorem cjne0d

Description: A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ⊢ φ → A ∈ ℂ
cjne0d.2 ⊢ φ → A ≠ 0
Assertion cjne0d ⊢ φ → A ‾ ≠ 0

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ φ → A ∈ ℂ
2 cjne0d.2 ⊢ φ → A ≠ 0
3 cjne0 ⊢ A ∈ ℂ → A ≠ 0 ↔ A ‾ ≠ 0
4 1 3 syl ⊢ φ → A ≠ 0 ↔ A ‾ ≠ 0
5 2 4 mpbid ⊢ φ → A ‾ ≠ 0