Description: Hypothesis for weak deduction theorem to eliminate A =/= 0 . (Contributed by NM, 15-May-1999)
Ref | Expression | ||
---|---|---|---|
Assertion | elimne0 | ⊢ if ( 𝐴 ≠ 0 , 𝐴 , 1 ) ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 | ⊢ ( 𝐴 = if ( 𝐴 ≠ 0 , 𝐴 , 1 ) → ( 𝐴 ≠ 0 ↔ if ( 𝐴 ≠ 0 , 𝐴 , 1 ) ≠ 0 ) ) | |
2 | neeq1 | ⊢ ( 1 = if ( 𝐴 ≠ 0 , 𝐴 , 1 ) → ( 1 ≠ 0 ↔ if ( 𝐴 ≠ 0 , 𝐴 , 1 ) ≠ 0 ) ) | |
3 | ax-1ne0 | ⊢ 1 ≠ 0 | |
4 | 1 2 3 | elimhyp | ⊢ if ( 𝐴 ≠ 0 , 𝐴 , 1 ) ≠ 0 |