Theorem breqtrrid 4163

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 2244	. 2 |- ( ph -> B = C )
4	1, 3	breqtrid 4162	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1402   class class class wbr 4125

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126

This theorem is referenced by:  xsubge0  10262  xposdif  10263  bernneq  11076  bitsfzo  12700  bitsmod  12701  bitsinv1lem  12706  pcge0  13070  ballotfilem5  13220  rpabscxpbnd  15965  lgsdir2lem2  16062  2lgsoddprmlem3  16144  eupth2lem3lem3fi  16625  eupth2lembfi  16632  trilpolemclim  16990  trilpolemlt1  16995  nconstwlpolemgt0  17019