Theorem ertr3d 6491

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
ersymb.1	|- ( ph -> R Er X )
ertr3d.5	|- ( ph -> B R A )
ertr3d.6	|- ( ph -> B R C )

Assertion

Ref	Expression
ertr3d	|- ( ph -> A R C )

Proof of Theorem ertr3d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 |- ( ph -> R Er X )
2		ertr3d.5	. . 3 |- ( ph -> B R A )
3	1, 2	ersym 6485	. 2 |- ( ph -> A R B )
4		ertr3d.6	. 2 |- ( ph -> B R C )
5	1, 3, 4	ertrd 6489	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 3965   Er wer 6470

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-er 6473

This theorem is referenced by:  xmetresbl  12800