Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs, and conversely. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 11-Jul-2020) Strengthen to a biconditional. (Revised by BJ, 10-Aug-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | had1 | ⊢ ( 𝜑 ↔ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ↔ 𝜒 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hadrot | ⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ hadd ( 𝜓 , 𝜒 , 𝜑 ) ) | |
| 2 | hadbi | ⊢ ( hadd ( 𝜓 , 𝜒 , 𝜑 ) ↔ ( ( 𝜓 ↔ 𝜒 ) ↔ 𝜑 ) ) | |
| 3 | 1 2 | bitri | ⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜓 ↔ 𝜒 ) ↔ 𝜑 ) ) |
| 4 | birot | ⊢ ( ( 𝜑 ↔ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ↔ 𝜒 ) ) ) ↔ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜓 ↔ 𝜒 ) ↔ 𝜑 ) ) ) | |
| 5 | 3 4 | mpbir | ⊢ ( 𝜑 ↔ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜓 ↔ 𝜒 ) ) ) |