Theorem syl6eqbr 2652

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 2629	. 2 |- (ph -> (ARC <-> BRC))
4	1, 3	mpbiri 194	1 |- (ph -> ARC)

Syntax hints:   -> wi 3   = wceq 956   class class class wbr 2619

This theorem is referenced by:  syl6eqbrr 2653  mapdom2 4494  unifiOLD 4557  fodomfiOLD 4566  pm54.43 4572  expmwordit 6606  exple1t 6607  seq1bnd 6910  facwordit 6944  faclbnd3 6947  bcpasc 6969  efcltlem2 7305  ruclem27 7536  nmosetn0 8428  nmo0 8451  siii 8513  bcsALT 9046  occllem5 9177  branmfnt 10038

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620

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