Theorem necon3d 2249

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 2247	. 2 |- ( ph -> ( C =/= D -> -. A = B ) )
3		df-ne 2206	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 151	1 |- ( ph -> ( C =/= D -> A =/= B ) )

Syntax hints:   -. wn 3   -> 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

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

This theorem is referenced by:  necon3i  2253  pm13.18  2286  ssn0  3259  suppssfv  5708  suppssov1  5709  nnmord  6090  findcard2  6346  findcard2s  6347  nn0n0n1ge2  8311