Theorem necon3d 2289

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 2287	. 2 |- ( ph -> ( C =/= D -> -. A = B ) )
3		df-ne 2246	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 160	1 |- ( ph -> ( C =/= D -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by:  necon3i  2293  pm13.18  2326  ssn0  3286  suppssfv  5728  suppssov1  5729  nnmord  6113  findcard2  6373  findcard2s  6374  addn0nid  7478  nn0n0n1ge2  8418  divgcdcoprmex  10484