Theorem elimne0 11195

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 3026	. 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2		neeq1 3026	. 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3		ax-1ne0 11168	. 2 ⊢ 1 ≠ 0
4	1, 2, 3	elimhyp 4558	1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

Syntax hints:   ≠ wne 2964  ifcif 4492  0cc0 11099  1c1 11100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-1ne0 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-if 4493

This theorem is referenced by:  sqdivzi  36118