Theorem ertr3d 6398

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 6392	. 2 |- ( ph -> A R B )
4		ertr3d.6	. 2 |- ( ph -> B R C )
5	1, 3, 4	ertrd 6396	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 3893   Er wer 6377

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-er 6380

This theorem is referenced by:  xmetresbl  12422