This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem annotanannot

Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022) (Proof shortened by Wolf Lammen, 1-Apr-2022)

Ref Expression
Assertion annotanannot ⊢ φ ∧ ¬ φ ∧ ψ ↔ φ ∧ ¬ ψ

Proof

Step Hyp Ref Expression
1 ibar ⊢ φ → ψ ↔ φ ∧ ψ
2 1 bicomd ⊢ φ → φ ∧ ψ ↔ ψ
3 2 notbid ⊢ φ → ¬ φ ∧ ψ ↔ ¬ ψ
4 3 pm5.32i ⊢ φ ∧ ¬ φ ∧ ψ ↔ φ ∧ ¬ ψ