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Theorem ertr2d 6922
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
ertr2d  |-  ( ph  ->  C R A )

Proof of Theorem ertr2d
StepHypRef Expression
1 ersymb.1 . 2  |-  ( ph  ->  R  Er  X )
2 ertrd.5 . . 3  |-  ( ph  ->  A R B )
3 ertrd.6 . . 3  |-  ( ph  ->  B R C )
41, 2, 3ertrd 6921 . 2  |-  ( ph  ->  A R C )
51, 4ersym 6917 1  |-  ( ph  ->  C R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4212    Er wer 6902
This theorem is referenced by:  pi1xfrcnvlem  19081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-er 6905
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