Theorem breqtrid 3973

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 3962	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1332   class class class wbr 3937

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938

This theorem is referenced by:  breqtrrid  3974  phplem3  6756