Theorem breqtrid 4019

Description: B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Hypotheses

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

Assertion

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

Proof of Theorem breqtrid

Step	Hyp	Ref	Expression
1		breqtrid.1	. . 3 |- A R B
2	1	a1i 9	. 2 |- ( ph -> A R B )
3		breqtrid.2	. 2 |- ( ph -> B = C )
4	2, 3	breqtrd 4008	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1343   class class class wbr 3982

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983

This theorem is referenced by:  breqtrrid  4020  phplem3  6820