Theorem syl5eqbr 3838

Description: B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Hypotheses

Ref	Expression
syl5eqbr.1	|- A = B
syl5eqbr.2	|- ( ph -> B R C )

Assertion

Ref	Expression
syl5eqbr	|- ( ph -> A R C )

Proof of Theorem syl5eqbr

Step	Hyp	Ref	Expression
1		syl5eqbr.2	. 2 |- ( ph -> B R C )
2		syl5eqbr.1	. 2 |- A = B
3		eqid 2083	. 2 |- C = C
4	1, 2, 3	3brtr4g 3837	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1285   class class class wbr 3805

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806

This theorem is referenced by:  xp1en  6389  caucvgprlemm  6990  intqfrac2  9471  m1modge3gt1  9523  bernneq2  9761  nno  10531  oddprmge3  10741  sqnprm  10742  oddennn  10830