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Theorem elimne0 10608
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 3069 . 2 (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2 neeq1 3069 . 2 (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3 ax-1ne0 10583 . 2 1 ≠ 0
41, 2, 3elimhyp 4503 1 if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Colors of variables: wff setvar class
Syntax hints:  wne 3007  ifcif 4440  0cc0 10514  1c1 10515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793  ax-1ne0 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ne 3008  df-if 4441
This theorem is referenced by:  sqdivzi  32967
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