Theorem elimne0 10030

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 2856	. 2 |- ( A = if ( A =/= 0 , A , 1 ) -> ( A =/= 0 <-> if ( A =/= 0 , A , 1 ) =/= 0 ) )
2		neeq1 2856	. 2 |- ( 1 = if ( A =/= 0 , A , 1 ) -> ( 1 =/= 0 <-> if ( A =/= 0 , A , 1 ) =/= 0 ) )
3		ax-1ne0 10005	. 2 |- 1 =/= 0
4	1, 2, 3	elimhyp 4146	1 |- if ( A =/= 0 , A , 1 ) =/= 0

Syntax hints:   =/= wne 2794   ifcif 4086   0cc0 9936   1c1 9937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-1ne0 10005

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-if 4087

This theorem is referenced by:  sqdivzi  31610