Theorem syl6eqbr 2642

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 2619	. 2 |- (ph -> (ARC <-> BRC))
4	1, 3	mpbiri 194	1 |- (ph -> ARC)

Syntax hints:   -> wi 3   = wceq 953   class class class wbr 2609

This theorem is referenced by:  syl6eqbrr 2643  mapdom2 4474  unifi 4532  fodomfi 4540  pm54.43 4546  expmwordit 6537  exple1t 6538  seq1bnd 6847  facwordit 6881  faclbnd3 6884  bcpasc 6907  efcltlem2 7247  ruclem27 7479  nmosetn0 8360  nmo0 8383  siii 8444  bcsALT 8967  occllem5 9093  branmfnt 9951

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610

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