Theorem eqnetrrd 2390

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
eqnetrrd.1	|- ( ph -> A = B )
eqnetrrd.2	|- ( ph -> A =/= C )

Assertion

Ref	Expression
eqnetrrd	|- ( ph -> B =/= C )

Proof of Theorem eqnetrrd

Step	Hyp	Ref	Expression
1		eqnetrrd.1	. . 3 |- ( ph -> A = B )
2	1	eqcomd 2199	. 2 |- ( ph -> B = A )
3		eqnetrrd.2	. 2 |- ( ph -> A =/= C )
4	2, 3	eqnetrd 2388	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   = wceq 1364   =/= wne 2364

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ne 2365

This theorem is referenced by:  netap  7304  2omotaplemap  7307  pcadd  12468