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

Step	Hyp	Ref	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
4	1, 2, 3	elimhyp 3955	1 |- if ( A =/= 0 , A , 1 ) =/= 0

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