Theorem nbrne1 4037

Description: Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Assertion

Ref	Expression
nbrne1	|- ( ( A R B /\ -. A R C ) -> B =/= C )

Proof of Theorem nbrne1

Step	Hyp	Ref	Expression
1		breq2 4022	. . . 4 |- ( B = C -> ( A R B <-> A R C ) )
2	1	biimpcd 159	. . 3 |- ( A R B -> ( B = C -> A R C ) )
3	2	necon3bd 2403	. 2 |- ( A R B -> ( -. A R C -> B =/= C ) )
4	3	imp 124	1 |- ( ( A R B /\ -. A R C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   =/= wne 2360   class class class wbr 4018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019

This theorem is referenced by:  zeneo  11908