Theorem breqtrrid 4043

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 2183	. 2 |- ( ph -> B = C )
4	1, 3	breqtrid 4042	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1353   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by:  xsubge0  9884  xposdif  9885  bernneq  10644  pcge0  12315  rpabscxpbnd  14499  lgsdir2lem2  14570  2lgsoddprmlem3  14599  trilpolemclim  14925  trilpolemlt1  14930  nconstwlpolemgt0  14953