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