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Theorem ertr4d 7322
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 7315 . 2  |-  ( ph  ->  B R C )
51, 2, 4ertrd 7319 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4439    Er wer 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-er 7303
This theorem is referenced by:  erref  7323  erdisj  7351  nqereu  9296  nqereq  9302  efgredeu  16969  pi1xfr  21721  pi1xfrcnvlem  21722
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