Description: When the branches are not equal, an "if" condition results in the first branch if and only if its condition is true. (Contributed by SN, 16-Oct-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | iftrueb | ⊢ ( 𝐴 ≠ 𝐵 → ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ↔ 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | necom | ⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 ) | |
| 2 | 1 | biimpi | ⊢ ( 𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴 ) |
| 3 | iffalse | ⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) | |
| 4 | 3 | neeq1d | ⊢ ( ¬ 𝜑 → ( if ( 𝜑 , 𝐴 , 𝐵 ) ≠ 𝐴 ↔ 𝐵 ≠ 𝐴 ) ) |
| 5 | 2 4 | syl5ibrcom | ⊢ ( 𝐴 ≠ 𝐵 → ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) ≠ 𝐴 ) ) |
| 6 | 5 | necon4bd | ⊢ ( 𝐴 ≠ 𝐵 → ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 → 𝜑 ) ) |
| 7 | iftrue | ⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) | |
| 8 | 6 7 | impbid1 | ⊢ ( 𝐴 ≠ 𝐵 → ( if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ↔ 𝜑 ) ) |