Theorem elimne0 11113

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 2991	. 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2		neeq1 2991	. 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3		ax-1ne0 11086	. 2 ⊢ 1 ≠ 0
4	1, 2, 3	elimhyp 4542	1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

Syntax hints:   ≠ wne 2929  ifcif 4476  0cc0 11017  1c1 11018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-1ne0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-if 4477

This theorem is referenced by:  sqdivzi  35844