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Theorem ertr3d 7326
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1  |-  ( ph  ->  R  Er  X )
ertr3d.5  |-  ( ph  ->  B R A )
ertr3d.6  |-  ( ph  ->  B R C )
Assertion
Ref Expression
ertr3d  |-  ( ph  ->  A R C )

Proof of Theorem ertr3d
StepHypRef Expression
1 ersymb.1 . 2  |-  ( ph  ->  R  Er  X )
2 ertr3d.5 . . 3  |-  ( ph  ->  B R A )
31, 2ersym 7320 . 2  |-  ( ph  ->  A R B )
4 ertr3d.6 . 2  |-  ( ph  ->  B R C )
51, 3, 4ertrd 7324 1  |-  ( ph  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4447    Er wer 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-er 7308
This theorem is referenced by:  nqereq  9309  efgred2  16567  xmetresbl  20675  pcophtb  21264  pi1xfr  21290  pi1xfrcnvlem  21291  prtlem10  30210
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