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Metamath Proof Explorer

Theorem necon3bbid

Description: Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007)

Ref Expression
Hypothesis necon3bbid.1 ⊢ φ → ψ ↔ A = B
Assertion necon3bbid ⊢ φ → ¬ ψ ↔ A ≠ B

Proof

Step Hyp Ref Expression
1 necon3bbid.1 ⊢ φ → ψ ↔ A = B
2 1 bicomd ⊢ φ → A = B ↔ ψ
3 2 necon3abid ⊢ φ → A ≠ B ↔ ¬ ψ
4 3 bicomd ⊢ φ → ¬ ψ ↔ A ≠ B