Theorem eqnetrrd 2446

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 2244	. 2 |- ( ph -> B = A )
3		eqnetrrd.2	. 2 |- ( ph -> A =/= C )
4	2, 3	eqnetrd 2444	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   = wceq 1402   =/= wne 2420

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is referenced by:  netap  7610  2omotaplemap  7613  pcadd  13097