Theorem elimne0 9368

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 2611	. 2 |- ( A = if ( A =/= 0 , A , 1 ) -> ( A =/= 0 <-> if ( A =/= 0 , A , 1 ) =/= 0 ) )
2		neeq1 2611	. 2 |- ( 1 = if ( A =/= 0 , A , 1 ) -> ( 1 =/= 0 <-> if ( A =/= 0 , A , 1 ) =/= 0 ) )
3		ax-1ne0 9343	. 2 |- 1 =/= 0
4	1, 2, 3	elimhyp 3843	1 |- if ( A =/= 0 , A , 1 ) =/= 0

Syntax hints:   =/= wne 2601   ifcif 3786   0cc0 9274   1c1 9275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-1ne0 9343

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-if 3787

This theorem is referenced by:  sqdivzi  27352