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Theorem ertr2d 6693
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 6692 . 2  |-  ( ph  ->  A R C )
51, 4ersym 6688 1  |-  ( ph  ->  C R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4039    Er wer 6673
This theorem is referenced by:  pi1xfrcnvlem  18570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-er 6676
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