Theorem ertrd 6145

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 6144	. 2 |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )
5	1, 2, 4	mp2and 423	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 3785   Er wer 6126

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-co 4372  df-er 6129

This theorem is referenced by:  ertr2d  6146  ertr3d  6147  ertr4d  6148  erinxp  6203