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Theorem ertr3d 6678
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 6672 . 2  |-  ( ph  ->  A R B )
4 ertr3d.6 . 2  |-  ( ph  ->  B R C )
51, 3, 4ertrd 6676 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4023    Er wer 6657
This theorem is referenced by:  nqereq  8559  efgred2  15062  xmetresbl  17983  pcophtb  18527  pi1xfr  18553  pi1xfrcnvlem  18554  prtlem10  26733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-er 6660
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