Theorem breqtrrid 4121

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 2235	. 2 |- ( ph -> B = C )
4	1, 3	breqtrid 4120	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1395   class class class wbr 4083

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  xsubge0  10077  xposdif  10078  bernneq  10882  bitsfzo  12466  bitsmod  12467  bitsinv1lem  12472  pcge0  12836  rpabscxpbnd  15614  lgsdir2lem2  15708  2lgsoddprmlem3  15790  trilpolemclim  16404  trilpolemlt1  16409  nconstwlpolemgt0  16432