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Theorem ertrd 6209
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
ertrd  |-  ( ph  ->  A R C )

Proof of Theorem ertrd
StepHypRef Expression
1 ertrd.5 . 2  |-  ( ph  ->  A R B )
2 ertrd.6 . 2  |-  ( ph  ->  B R C )
3 ersymb.1 . . 3  |-  ( ph  ->  R  Er  X )
43ertr 6208 . 2  |-  ( ph  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
51, 2, 4mp2and 424 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   class class class wbr 3805    Er wer 6190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-co 4400  df-er 6193
This theorem is referenced by:  ertr2d  6210  ertr3d  6211  ertr4d  6212  erinxp  6267
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