Description: Theorem *5.75 of WhiteheadRussell p. 126. (Contributed by NM, 3-Jan-2005) (Proof shortened by Andrew Salmon, 7-May-2011) (Proof shortened by Wolf Lammen, 23-Dec-2012) (Proof shortened by Kyle Wyonch, 12-Feb-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | pm5.75 | ⊢ ( ( ( 𝜒 → ¬ 𝜓 ) ∧ ( 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) ) → ( ( 𝜑 ∧ ¬ 𝜓 ) ↔ 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | anbi1 | ⊢ ( ( 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) → ( ( 𝜑 ∧ ¬ 𝜓 ) ↔ ( ( 𝜓 ∨ 𝜒 ) ∧ ¬ 𝜓 ) ) ) | |
| 2 | biorf | ⊢ ( ¬ 𝜓 → ( 𝜒 ↔ ( 𝜓 ∨ 𝜒 ) ) ) | |
| 3 | 2 | bicomd | ⊢ ( ¬ 𝜓 → ( ( 𝜓 ∨ 𝜒 ) ↔ 𝜒 ) ) |
| 4 | 3 | pm5.32ri | ⊢ ( ( ( 𝜓 ∨ 𝜒 ) ∧ ¬ 𝜓 ) ↔ ( 𝜒 ∧ ¬ 𝜓 ) ) |
| 5 | 1 4 | bitrdi | ⊢ ( ( 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) → ( ( 𝜑 ∧ ¬ 𝜓 ) ↔ ( 𝜒 ∧ ¬ 𝜓 ) ) ) |
| 6 | abai | ⊢ ( ( 𝜒 ∧ ¬ 𝜓 ) ↔ ( 𝜒 ∧ ( 𝜒 → ¬ 𝜓 ) ) ) | |
| 7 | 6 | rbaib | ⊢ ( ( 𝜒 → ¬ 𝜓 ) → ( ( 𝜒 ∧ ¬ 𝜓 ) ↔ 𝜒 ) ) |
| 8 | 5 7 | sylan9bbr | ⊢ ( ( ( 𝜒 → ¬ 𝜓 ) ∧ ( 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) ) → ( ( 𝜑 ∧ ¬ 𝜓 ) ↔ 𝜒 ) ) |