Theorem ertr3d 6637

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 6631	. 2 |- ( ph -> A R B )
4		ertr3d.6	. 2 |- ( ph -> B R C )
5	1, 3, 4	ertrd 6635	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 4043   Er wer 6616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-er 6619

This theorem is referenced by:  xmetresbl  14854