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 ) )