Theorem breqtrrdi 4029

Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

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

Assertion

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

Proof of Theorem breqtrrdi

Step	Hyp	Ref	Expression
1		breqtrrdi.1	. 2 |- ( ph -> A R B )
2		breqtrrdi.2	. . 3 |- C = B
3	2	eqcomi 2174	. 2 |- B = C
4	1, 3	breqtrdi 4028	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1348   class class class wbr 3987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988

This theorem is referenced by:  enpr2d  6793  fiunsnnn  6857  unsnfi  6894  eninl  7072  eninr  7073  difinfinf  7076  exmidfodomrlemr  7172  exmidfodomrlemrALT  7173  dju1en  7183  djucomen  7186  djuassen  7187  xpdjuen  7188  gtndiv  9300  intqfrac2  10268  uzenom  10374  xrmaxiflemval  11206  ege2le3  11627  eirraplem  11732  pcprendvds  12237  pcpremul  12240  pcfaclem  12294  infpnlem2  12305  2strstr1g  12514  lmcn2  13039  dveflem  13446  tangtx  13518  ioocosf1o  13534  lgsdirprm  13694  sbthom  14023  nconstwlpolemgt0  14060