Theorem necon3d 2554

Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Hypothesis

Ref	Expression
necon3d.1	|- (ph -> (A = B -> C = D))

Assertion

Ref	Expression
necon3d	|- (ph -> (C =/= D -> A =/= B))

Proof of Theorem necon3d

Step	Hyp	Ref	Expression
1		necon3d.1	. . 3 |- (ph -> (A = B -> C = D))
2	1	necon3ad 2552	. 2 |- (ph -> (C =/= D -> -. A = B))
3		df-ne 2518	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl6ibr 218	1 |- (ph -> (C =/= D -> A =/= B))

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-ne 2518

This theorem is referenced by:  necon3i  2555  pm13.18  2588  ssn0  3583  pssdifn0  3611  uniintsn  3963  evenodddisj  4516  sfinltfin  4535  leltctr  6212  nnltp1c  6262