Theorem breqtrrid 4168

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 4167	1 |- ( ph -> A R C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   class class class wbr 4130

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  xsubge0  10294  xposdif  10295  bernneq  11113  bitsfzo  12741  bitsmod  12742  bitsinv1lem  12747  pcge0  13115  ballotfilem5  13294  rpabscxpbnd  16137  lgsdir2lem2  16314  2lgsoddprmlem3  16396  eupth2lem3lem3fi  16877  eupth2lembfi  16884  trilpolemclim  17252  trilpolemlt1  17257  nconstwlpolemgt0  17281