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  6795  findcard2  7199  findcard2s  7200  addn0nid  8707  nn0n0n1ge2  9724  xnegdi  10286  efne0  12469  divgcdcoprmex  12904  pceulem  13101  pcqmul  13110  pcqcl  13113  pcaddlem  13146  pcadd  13147  grpinvnz  13934  ringelnzr  14602  lmodfopne  14771  lmodindp1  14873  birthdaylem1g  16242  clwwlkccat  16864  clwwlknonel  16895