Theorem elimgt0

Description: Hypothesis for weak deduction theorem to eliminate 0 < A . (Contributed by NM, 15-May-1999)

	Ref	Expression
Assertion	elimgt0	⊢ 0 < if ( 0 < 𝐴 , 𝐴 , 1 )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝐴 = if ( 0 < 𝐴 , 𝐴 , 1 ) → ( 0 < 𝐴 ↔ 0 < if ( 0 < 𝐴 , 𝐴 , 1 ) ) )
2		breq2	⊢ ( 1 = if ( 0 < 𝐴 , 𝐴 , 1 ) → ( 0 < 1 ↔ 0 < if ( 0 < 𝐴 , 𝐴 , 1 ) ) )
3		0lt1	⊢ 0 < 1
4	1 2 3	elimhyp	⊢ 0 < if ( 0 < 𝐴 , 𝐴 , 1 )