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Theorem ertr3d 6190
 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 6184 . 2 (𝜑𝐴𝑅𝐵)
4 ertr3d.6 . 2 (𝜑𝐵𝑅𝐶)
51, 3, 4ertrd 6188 1 (𝜑𝐴𝑅𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   class class class wbr 3793   Er wer 6169 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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-er 6172 This theorem is referenced by: (None)
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