Metamath Proof Explorer


Theorem elimne0

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

Proof

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