Theorem eqnetrd 2360

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
eqnetrd.1	|- ( ph -> A = B )
eqnetrd.2	|- ( ph -> B =/= C )

Assertion

Ref	Expression
eqnetrd	|- ( ph -> A =/= C )

Proof of Theorem eqnetrd

Step	Hyp	Ref	Expression
1		eqnetrd.2	. 2 |- ( ph -> B =/= C )
2		eqnetrd.1	. . 3 |- ( ph -> A = B )
3	2	neeq1d 2354	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
4	1, 3	mpbird 166	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1343   =/= wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2337

This theorem is referenced by:  eqnetrrd  2362  frecabcl  6367  frecsuclem  6374  omp1eomlem  7059  xaddnemnf  9793  xaddnepnf  9794  hashprg  10721  bezoutr1  11966  phibndlem  12148  dfphi2  12152  lgsne0  13579  2sqlem8a  13598  2sqlem8  13599