Theorem ne3anior 2396

Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)

Assertion

Ref	Expression
ne3anior	|- ( ( A =/= B /\ C =/= D /\ E =/= F ) <-> -. ( A = B \/ C = D \/ E = F ) )

Proof of Theorem ne3anior

Step	Hyp	Ref	Expression
1		df-ne 2309	. . 3 |- ( A =/= B <-> -. A = B )
2		df-ne 2309	. . 3 |- ( C =/= D <-> -. C = D )
3		df-ne 2309	. . 3 |- ( E =/= F <-> -. E = F )
4	1, 2, 3	3anbi123i 1170	. 2 |- ( ( A =/= B /\ C =/= D /\ E =/= F ) <-> ( -. A = B /\ -. C = D /\ -. E = F ) )
5		3ioran 977	. 2 |- ( -. ( A = B \/ C = D \/ E = F ) <-> ( -. A = B /\ -. C = D /\ -. E = F ) )
6	4, 5	bitr4i 186	1 |- ( ( A =/= B /\ C =/= D /\ E =/= F ) <-> -. ( A = B \/ C = D \/ E = F ) )

Syntax hints:   -. wn 3   <-> wb 104   \/ w3o 961   /\ w3a 962   = 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  ax-io 698

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-ne 2309

This theorem is referenced by: (None)