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Theorem ertr2d 7220
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
ertr2d  |-  ( ph  ->  C R A )

Proof of Theorem ertr2d
StepHypRef Expression
1 ersymb.1 . 2  |-  ( ph  ->  R  Er  X )
2 ertrd.5 . . 3  |-  ( ph  ->  A R B )
3 ertrd.6 . . 3  |-  ( ph  ->  B R C )
41, 2, 3ertrd 7219 . 2  |-  ( ph  ->  A R C )
51, 4ersym 7215 1  |-  ( ph  ->  C R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4392    Er wer 7200
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 4513  ax-nul 4521  ax-pr 4631
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 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-er 7203
This theorem is referenced by:  pi1xfrcnvlem  20746
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