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Theorem elimne0 8829
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 2454 . 2  |-  ( A  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( A  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
2 neeq1 2454 . 2  |-  ( 1  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( 1  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
3 ax-1ne0 8806 . 2  |-  1  =/=  0
41, 2, 3elimhyp 3613 1  |-  if ( A  =/=  0 ,  A ,  1 )  =/=  0
Colors of variables: wff set class
Syntax hints:    =/= wne 2446   ifcif 3565   0cc0 8737   1c1 8738
This theorem is referenced by:  sqdivzi  24079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-1ne0 8806
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-if 3566
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