Description: Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-animbi | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) | |
| 2 | pm3.4 | ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 → 𝜓 ) ) | |
| 3 | 1 2 | 2thd | ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) ) |
| 4 | biimp | ⊢ ( ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) → ( 𝜑 → ( 𝜑 → 𝜓 ) ) ) | |
| 5 | 4 | pm2.43d | ⊢ ( ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) |
| 6 | biimpr | ⊢ ( ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜑 ) ) | |
| 7 | 5 6 | mpd | ⊢ ( ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) → 𝜑 ) |
| 8 | 7 5 | jcai | ⊢ ( ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) → ( 𝜑 ∧ 𝜓 ) ) |
| 9 | 3 8 | impbii | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ↔ ( 𝜑 → 𝜓 ) ) ) |