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Theorem elimne0 11252
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 3002 . 2 (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
2 neeq1 3002 . 2 (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0))
3 ax-1ne0 11225 . 2 1 ≠ 0
41, 2, 3elimhyp 4590 1 if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Colors of variables: wff setvar class
Syntax hints:  wne 2939  ifcif 4524  0cc0 11156  1c1 11157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-1ne0 11225
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-if 4525
This theorem is referenced by:  sqdivzi  35729
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