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Theorem ertrd 6676
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
ertrd  |-  ( ph  ->  A R C )

Proof of Theorem ertrd
StepHypRef Expression
1 ertrd.5 . 2  |-  ( ph  ->  A R B )
2 ertrd.6 . 2  |-  ( ph  ->  B R C )
3 ersymb.1 . . 3  |-  ( ph  ->  R  Er  X )
43ertr 6675 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
51, 2, 4mp2and 660 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:  ertr2d  6677  ertr3d  6678  ertr4d  6679  erinxp  6733  nqereq  8559  adderpq  8580  mulerpq  8581  efgred2  15062  efgcpbllemb  15064  efgcpbl2  15066  pcophtb  18527  pi1xfr  18553  pi1xfrcnvlem  18554
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-co 4698  df-er 6660
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