Theorem ne3anior 2293

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 2206	. . 3 |- ( A =/= B <-> -. A = B )
2		df-ne 2206	. . 3 |- ( C =/= D <-> -. C = D )
3		df-ne 2206	. . 3 |- ( E =/= F <-> -. E = F )
4	1, 2, 3	3anbi123i 1093	. 2 |- ( ( A =/= B /\ C =/= D /\ E =/= F ) <-> ( -. A = B /\ -. C = D /\ -. E = F ) )
5		3ioran 900	. 2 |- ( -. ( A = B \/ C = D \/ E = F ) <-> ( -. A = B /\ -. C = D /\ -. E = F ) )
6	4, 5	bitr4i 176	1 |- ( ( A =/= B /\ C =/= D /\ E =/= F ) <-> -. ( A = B \/ C = D \/ E = F ) )

Syntax hints:   -. wn 3   <-> wb 98   \/ w3o 884   /\ w3a 885   = 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  ax-io 630

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-ne 2206

This theorem is referenced by: (None)