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Theorem ertrd 6288
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 6287 . 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 3837    Er wer 6269
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-co 4437  df-er 6272
This theorem is referenced by:  ertr2d  6289  ertr3d  6290  ertr4d  6291  erinxp  6346
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