Theorem breqtrrid 4019

Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

Ref	Expression
breqtrrid.1	|- A R B
breqtrrid.2	|- ( ph -> C = B )

Assertion

Ref	Expression
breqtrrid	|- ( ph -> A R C )

Proof of Theorem breqtrrid

Step	Hyp	Ref	Expression
1		breqtrrid.1	. 2 |- A R B
2		breqtrrid.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2171	. 2 |- ( ph -> B = C )
4	1, 3	breqtrid 4018	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1343   class class class wbr 3981

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982

This theorem is referenced by:  xsubge0  9813  xposdif  9814  bernneq  10571  pcge0  12240  rpabscxpbnd  13459  lgsdir2lem2  13530  trilpolemclim  13875  trilpolemlt1  13880  nconstwlpolemgt0  13902