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Theorem elimne0 9887
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 2843 . 2 (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2 neeq1 2843 . 2 (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3 ax-1ne0 9862 . 2 1 ≠ 0
41, 2, 3elimhyp 4095 1 if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Colors of variables: wff setvar class
Syntax hints:  wne 2779  ifcif 4035  0cc0 9793  1c1 9794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-1ne0 9862
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-ne 2781  df-if 4036
This theorem is referenced by:  sqdivzi  30697
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