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Theorem ertr3d 6451
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
ertr3d.5 (𝜑𝐵𝑅𝐴)
ertr3d.6 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
ertr3d (𝜑𝐴𝑅𝐶)

Proof of Theorem ertr3d
StepHypRef Expression
1 ersymb.1 . 2 (𝜑𝑅 Er 𝑋)
2 ertr3d.5 . . 3 (𝜑𝐵𝑅𝐴)
31, 2ersym 6445 . 2 (𝜑𝐴𝑅𝐵)
4 ertr3d.6 . 2 (𝜑𝐵𝑅𝐶)
51, 3, 4ertrd 6449 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   class class class wbr 3933   Er wer 6430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-er 6433
This theorem is referenced by:  xmetresbl  12639
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