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Theorem elimne0 9486
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 2732 . 2  |-  ( A  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( A  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
2 neeq1 2732 . 2  |-  ( 1  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( 1  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
3 ax-1ne0 9461 . 2  |-  1  =/=  0
41, 2, 3elimhyp 3955 1  |-  if ( A  =/=  0 ,  A ,  1 )  =/=  0
Colors of variables: wff setvar class
Syntax hints:    =/= wne 2647   ifcif 3898   0cc0 9392   1c1 9393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-1ne0 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-ne 2649  df-if 3899
This theorem is referenced by:  sqdivzi  27526
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