Theorem eqnetrd 2391

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 2385	. 2 |- ( ph -> ( A =/= C <-> B =/= C ) )
4	1, 3	mpbird 167	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1364   =/= wne 2367

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  eqnetrrd  2393  ifnetruedc  3603  ifnefals  3604  frecabcl  6466  frecsuclem  6473  omp1eomlem  7169  xaddnemnf  9949  xaddnepnf  9950  hashprg  10917  bezoutr1  12225  phibndlem  12409  dfphi2  12413  lgsne0  15363  2sqlem8a  15447  2sqlem8  15448