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Theorem ertr3d 6925
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
StepHypRef Expression
1 ersymb.1 . 2  |-  ( ph  ->  R  Er  X )
2 ertr3d.5 . . 3  |-  ( ph  ->  B R A )
31, 2ersym 6919 . 2  |-  ( ph  ->  A R B )
4 ertr3d.6 . 2  |-  ( ph  ->  B R C )
51, 3, 4ertrd 6923 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4214    Er wer 6904
This theorem is referenced by:  nqereq  8814  efgred2  15387  xmetresbl  18469  pcophtb  19056  pi1xfr  19082  pi1xfrcnvlem  19083  prtlem10  26716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-er 6907
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