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Theorem ertr3d 7124
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 7118 . 2  |-  ( ph  ->  A R B )
4 ertr3d.6 . 2  |-  ( ph  ->  B R C )
51, 3, 4ertrd 7122 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4297    Er wer 7103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-er 7106
This theorem is referenced by:  nqereq  9109  efgred2  16255  xmetresbl  20017  pcophtb  20606  pi1xfr  20632  pi1xfrcnvlem  20633  prtlem10  29015
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