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Theorem ertrd 7217
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 7216 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
51, 2, 4mp2and 679 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4390    Er wer 7198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-xp 4944  df-rel 4945  df-co 4947  df-er 7201
This theorem is referenced by:  ertr2d  7218  ertr3d  7219  ertr4d  7220  erinxp  7274  nqereq  9205  adderpq  9226  mulerpq  9227  efgred2  16354  efgcpbllemb  16356  efgcpbl2  16358  pcophtb  20717  pi1xfr  20743  pi1xfrcnvlem  20744
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