Theorem eqnetrd 2332

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 2326	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
4	1, 3	mpbird 166	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1331   =/= wne 2308

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-ne 2309

This theorem is referenced by:  eqnetrrd  2334  frecabcl  6296  frecsuclem  6303  omp1eomlem  6979  xaddnemnf  9640  xaddnepnf  9641  hashprg  10554  bezoutr1  11721  phibndlem  11892  dfphi2  11896