Theorem neeqtrd 2538

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 2530	. 2 |- (ph -> (A =/= B <-> A =/= C))
4	1, 3	mpbid 201	1 |- (ph -> A =/= C)

Syntax hints:   -> wi 4   = wceq 1642   =/= wne 2516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2518

This theorem is referenced by:  neeqtrrd  2540