Theorem neeqtrd 2375

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

Hypotheses

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

Assertion

Ref	Expression
neeqtrd	|- ( ph -> A =/= C )

Proof of Theorem neeqtrd

Step	Hyp	Ref	Expression
1		neeqtrd.1	. 2 |- ( ph -> A =/= B )
2		neeqtrd.2	. . 3 |- ( ph -> B = C )
3	2	neeq2d 2366	. 2 |- ( ph -> ( A =/= B <-> A =/= C ) )
4	1, 3	mpbid 147	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1353   =/= wne 2347

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-ne 2348

This theorem is referenced by:  neeqtrrd  2377  xaddass2  9866  modsumfzodifsn  10391