Theorem syl6eqbr 2725

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl6eqbr.1	|- (ph -> A = B)
syl6eqbr.2	|- BRC

Assertion

Ref	Expression
syl6eqbr	|- (ph -> ARC)

Proof of Theorem syl6eqbr

Step	Hyp	Ref	Expression
1		syl6eqbr.2	. 2 |- BRC
2		syl6eqbr.1	. . 3 |- (ph -> A = B)
3	2	breq1d 2702	. 2 |- (ph -> (ARC <-> BRC))
4	1, 3	mpbiri 192	1 |- (ph -> ARC)

Syntax hints:   -> wi 3   = wceq 992   class class class wbr 2692

This theorem is referenced by:  syl6eqbrr 2726  mapdom2 4641  fodomfi 4709  pm54.43 4715  expmwordi 6803  exple1 6804  seq1bndi 7113  facwordi 7147  faclbnd3 7150  bcpasci 7172  efcltlem2 7510  ruclem27 7748  nmosetn0 8682  nmo0 8706  siii 8769  bcsiALT 9322  occllem5 9453  branmfn 10317  branmfnOLD 10318  bfp 12065

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-sn 2470  df-pr 2471  df-op 2474  df-br 2693

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