Theorem eqnetrd 2351

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 2345	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
4	1, 3	mpbird 166	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1335   =/= wne 2327

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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-ne 2328

This theorem is referenced by:  eqnetrrd  2353  frecabcl  6340  frecsuclem  6347  omp1eomlem  7028  xaddnemnf  9743  xaddnepnf  9744  hashprg  10664  bezoutr1  11897  phibndlem  12068  dfphi2  12072