Theorem elimne0 9575

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 2735	. 2 |- ( A = if ( A =/= 0 , A , 1 ) -> ( A =/= 0 <-> if ( A =/= 0 , A , 1 ) =/= 0 ) )
2		neeq1 2735	. 2 |- ( 1 = if ( A =/= 0 , A , 1 ) -> ( 1 =/= 0 <-> if ( A =/= 0 , A , 1 ) =/= 0 ) )
3		ax-1ne0 9550	. 2 |- 1 =/= 0
4	1, 2, 3	elimhyp 3987	1 |- if ( A =/= 0 , A , 1 ) =/= 0

Syntax hints:   =/= wne 2649   ifcif 3929   0cc0 9481   1c1 9482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-1ne0 9550

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ne 2651  df-if 3930

This theorem is referenced by:  sqdivzi  29345