Theorem eqnetrd 2444

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 2438	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
4	1, 3	mpbird 167	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1402   =/= wne 2420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is referenced by:  eqnetrrd  2446  ifnetruedc  3681  ifnefals  3682  frecabcl  6660  frecsuclem  6667  omp1eomlem  7424  xaddnemnf  10238  xaddnepnf  10239  hashprg  11227  bezoutr1  12788  phibndlem  12972  dfphi2  12976  lgsne0  16071  2sqlem8a  16155  2sqlem8  16156