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Theorem elimne0 9074
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 2606 . 2  |-  ( A  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( A  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
2 neeq1 2606 . 2  |-  ( 1  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( 1  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
3 ax-1ne0 9051 . 2  |-  1  =/=  0
41, 2, 3elimhyp 3779 1  |-  if ( A  =/=  0 ,  A ,  1 )  =/=  0
Colors of variables: wff set class
Syntax hints:    =/= wne 2598   ifcif 3731   0cc0 8982   1c1 8983
This theorem is referenced by:  sqdivzi  25176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-1ne0 9051
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ne 2600  df-if 3732
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