Theorem syl6eqbr 3801

Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Hypotheses

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

Assertion

Ref	Expression
syl6eqbr	|- ( ph -> A R C )

Proof of Theorem syl6eqbr

Step	Hyp	Ref	Expression
1		syl6eqbr.2	. 2 |- B R C
2		syl6eqbr.1	. . 3 |- ( ph -> A = B )
3	2	breq1d 3774	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mpbiri 157	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1243   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by:  syl6eqbrr  3802  xltnegi  8748  leexp1a  9309  resqrexlemlo  9611