Theorem eqnetrd 2426

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 2420	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
4	1, 3	mpbird 167	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1397   =/= wne 2402

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 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-ne 2403

This theorem is referenced by:  eqnetrrd  2428  ifnetruedc  3649  ifnefals  3650  frecabcl  6564  frecsuclem  6571  omp1eomlem  7292  xaddnemnf  10091  xaddnepnf  10092  hashprg  11071  bezoutr1  12603  phibndlem  12787  dfphi2  12791  lgsne0  15766  2sqlem8a  15850  2sqlem8  15851