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

Proof of Theorem ertrd
StepHypRef Expression
1 ertrd.5 . 2 (𝜑𝐴𝑅𝐵)
2 ertrd.6 . 2 (𝜑𝐵𝑅𝐶)
3 ersymb.1 . . 3 (𝜑𝑅 Er 𝑋)
43ertr 6450 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 430 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   class class class wbr 3935   Er wer 6432
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 4052  ax-pow 4104  ax-pr 4137
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 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-xp 4551  df-rel 4552  df-co 4554  df-er 6435
This theorem is referenced by:  ertr2d  6452  ertr3d  6453  ertr4d  6454  erinxp  6509
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