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Theorem elimne0 9015
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 2558 . 2  |-  ( A  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( A  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
2 neeq1 2558 . 2  |-  ( 1  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( 1  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
3 ax-1ne0 8992 . 2  |-  1  =/=  0
41, 2, 3elimhyp 3730 1  |-  if ( A  =/=  0 ,  A ,  1 )  =/=  0
Colors of variables: wff set class
Syntax hints:    =/= wne 2550   ifcif 3682   0cc0 8923   1c1 8924
This theorem is referenced by:  sqdivzi  24963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-1ne0 8992
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-ne 2552  df-if 3683
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