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Theorem ertr4d 6766
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1  |-  ( ph  ->  R  Er  X )
ertr4d.5  |-  ( ph  ->  A R B )
ertr4d.6  |-  ( ph  ->  C R B )
Assertion
Ref Expression
ertr4d  |-  ( ph  ->  A R C )

Proof of Theorem ertr4d
StepHypRef Expression
1 ersymb.1 . 2  |-  ( ph  ->  R  Er  X )
2 ertr4d.5 . 2  |-  ( ph  ->  A R B )
3 ertr4d.6 . . 3  |-  ( ph  ->  C R B )
41, 3ersym 6759 . 2  |-  ( ph  ->  B R C )
51, 2, 4ertrd 6763 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 4104    Er wer 6744
This theorem is referenced by:  erref  6767  erdisj  6794  nqereu  8643  nqereq  8649  efgredeu  15160  pi1xfr  18657  pi1xfrcnvlem  18658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-er 6747
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