Theorem neeqtrd 2233

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 2224	. 2 |- ( ph -> ( A =/= B <-> A =/= C ) )
4	1, 3	mpbid 135	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206

This theorem is referenced by:  neeqtrrd  2235