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 ) ) |
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 ) ) |