Description: An alternate definition of the biconditional. Theorem *5.23 of WhiteheadRussell p. 124. (Contributed by NM, 27-Jun-2002) (Proof shortened by Wolf Lammen, 3-Nov-2013) (Proof shortened by NM, 29-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dfbi3 | ⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | con34b | ⊢ ( ( 𝜓 → 𝜑 ) ↔ ( ¬ 𝜑 → ¬ 𝜓 ) ) | |
| 2 | 1 | anbi2i | ⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → ¬ 𝜓 ) ) ) |
| 3 | dfbi2 | ⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜑 ) ) ) | |
| 4 | cases2 | ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → ¬ 𝜓 ) ) ) | |
| 5 | 2 3 4 | 3bitr4i | ⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ) ) |