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Theorem ertr3d 7647
 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 7641 . 2 (𝜑𝐴𝑅𝐵)
4 ertr3d.6 . 2 (𝜑𝐵𝑅𝐶)
51, 3, 4ertrd 7645 1 (𝜑𝐴𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   class class class wbr 4583   Er wer 7626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-er 7629 This theorem is referenced by:  nqereq  9636  efgred2  17989  xmetresbl  22052  pcophtb  22637  pi1xfr  22663  pi1xfrcnvlem  22664  erbr3b  28807  prtlem10  33168
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