Theorem neanior 1639

Description: A DeMorgan's law for inequality.

Assertion

Ref	Expression
neanior	|- ((A =/= B /\ C =/= D) <-> -. (A = B \/ C = D))

Proof of Theorem neanior

Step	Hyp	Ref	Expression
1		df-ne 1587	. . 3 |- (A =/= B <-> -. A = B)
2		df-ne 1587	. . 3 |- (C =/= D <-> -. C = D)
3	1, 2	anbi12i 482	. 2 |- ((A =/= B /\ C =/= D) <-> (-. A = B /\ -. C = D))
4		pm4.56 311	. 2 |- ((-. A = B /\ -. C = D) <-> -. (A = B \/ C = D))
5	3, 4	bitr 173	1 |- ((A =/= B /\ C =/= D) <-> -. (A = B \/ C = D))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   =/= wne 1585

This theorem is referenced by:  om00 4206  om00el 4207  muln0bt 5697  nmlno0lem 8453  nmlnop0ALT 9920  irred 10321  mdsym 10338  neiopne 10474

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ne 1587

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