Theorem ertrd 6517

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

Hypotheses

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

Assertion

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

Proof of Theorem ertrd

Step	Hyp	Ref	Expression
1		ertrd.5	. 2 |- ( ph -> A R B )
2		ertrd.6	. 2 |- ( ph -> B R C )
3		ersymb.1	. . 3 |- ( ph -> R Er X )
4	3	ertr 6516	. 2 |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )
5	1, 2, 4	mp2and 430	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 3982   Er wer 6498

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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-co 4613  df-er 6501

This theorem is referenced by:  ertr2d  6518  ertr3d  6519  ertr4d  6520  erinxp  6575