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 ∧ ¬ ( 𝜑𝜓 ) ) ↔ ( 𝜑 ∧ ¬ 𝜓 ) )

Proof

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