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Theorem ertrd 6692
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 6691 . 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 4039    Er wer 6673
This theorem is referenced by:  ertr2d  6693  ertr3d  6694  ertr4d  6695  erinxp  6749  nqereq  8575  adderpq  8596  mulerpq  8597  efgred2  15078  efgcpbllemb  15080  efgcpbl2  15082  pcophtb  18543  pi1xfr  18569  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-co 4714  df-er 6676
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