Theorem elimne0 11280

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 3009	. 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2		neeq1 3009	. 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3		ax-1ne0 11253	. 2 ⊢ 1 ≠ 0
4	1, 2, 3	elimhyp 4613	1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

Syntax hints:   ≠ wne 2946  ifcif 4548  0cc0 11184  1c1 11185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-1ne0 11253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-if 4549

This theorem is referenced by:  sqdivzi  35690