Theorem elimne0 11306

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

Assertion

Ref	Expression
elimne0	⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

Proof of Theorem elimne0

Step	Hyp	Ref	Expression
1		neeq1 3020	. 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2		neeq1 3020	. 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3		ax-1ne0 11279	. 2 ⊢ 1 ≠ 0
4	1, 2, 3	elimhyp 4550	1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

This proof depends on syntax axioms:   ≠ wne 2958  ifcif 4484  0cc0 11210  1c1 11211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-1ne0 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-if 4485

This theorem is used by:  sqdivzi  36620