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Theorem elimne0 11249
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
StepHypRef Expression
1 neeq1 3001 . 2 (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2 neeq1 3001 . 2 (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3 ax-1ne0 11222 . 2 1 ≠ 0
41, 2, 3elimhyp 4596 1 if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Colors of variables: wff setvar class
Syntax hints:  wne 2938  ifcif 4531  0cc0 11153  1c1 11154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-1ne0 11222
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-if 4532
This theorem is referenced by:  sqdivzi  35708
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