Theorem ne3anior 2603

Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)

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		3anor 948	. 2 |- ((A =/= B /\ C =/= D /\ E =/= F) <-> -. (-. A =/= B \/ -. C =/= D \/ -. E =/= F))
2		nne 2521	. . 3 |- (-. A =/= B <-> A = B)
3		nne 2521	. . 3 |- (-. C =/= D <-> C = D)
4		nne 2521	. . 3 |- (-. E =/= F <-> E = F)
5	2, 3, 4	3orbi123i 1141	. 2 |- ((-. A =/= B \/ -. C =/= D \/ -. E =/= F) <-> (A = B \/ C = D \/ E = F))
6	1, 5	xchbinx 301	1 |- ((A =/= B /\ C =/= D /\ E =/= F) <-> -. (A = B \/ C = D \/ E = F))

This proof depends on syntax axioms:  -. wn 3   <-> wb 176   \/ w3o 933   /\ w3a 934   = wceq 1642   =/= wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-ne 2519

This theorem is used by: (None)