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Theorem ertr2d 7365
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 7364 . 2  |-  ( ph  ->  A R C )
51, 4ersym 7360 1  |-  ( ph  ->  C R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   class class class wbr 4395    Er wer 7345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-er 7348
This theorem is referenced by:  pi1xfrcnvlem  21848
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