Theorem necon3d 2464

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 2462	. 2 |- ( ph -> ( C =/= D -> -. A = B ) )
3		df-ne 2421	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( C =/= D -> A =/= B ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  necon3i  2468  pm13.18  2501  ssn0  3566  suppssov1  6302  suppfnss  6501  suppssfvg  6507  nnmord  6794  findcard2  7197  findcard2s  7198  addn0nid  8707  nn0n0n1ge2  9726  xnegdi  10287  efne0  12470  divgcdcoprmex  12905  pceulem  13102  pcqmul  13111  pcqcl  13114  pcaddlem  13147  pcadd  13148  grpinvnz  13935  ringelnzr  14603  lmodfopne  14772  lmodindp1  14874  birthdaylem1g  16225  clwwlkccat  16847  clwwlknonel  16878