Theorem necon3d 2352

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 2350	. 2 |- ( ph -> ( C =/= D -> -. A = B ) )
3		df-ne 2309	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 161	1 |- ( ph -> ( C =/= D -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   =/= wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2309

This theorem is referenced by:  necon3i  2356  pm13.18  2389  ssn0  3405  suppssfv  5978  suppssov1  5979  nnmord  6413  findcard2  6783  findcard2s  6784  addn0nid  8136  nn0n0n1ge2  9121  xnegdi  9651  efne0  11384  divgcdcoprmex  11783