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Theorem elimne0 8845
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 2467 . 2  |-  ( A  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( A  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
2 neeq1 2467 . 2  |-  ( 1  =  if ( A  =/=  0 ,  A ,  1 )  -> 
( 1  =/=  0  <->  if ( A  =/=  0 ,  A ,  1 )  =/=  0 ) )
3 ax-1ne0 8822 . 2  |-  1  =/=  0
41, 2, 3elimhyp 3626 1  |-  if ( A  =/=  0 ,  A ,  1 )  =/=  0
Colors of variables: wff set class
Syntax hints:    =/= wne 2459   ifcif 3578   0cc0 8753   1c1 8754
This theorem is referenced by:  sqdivzi  24094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-1ne0 8822
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-if 3579
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