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Theorem ertr3d 8105
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 8099 . 2 (𝜑𝐴𝑅𝐵)
4 ertr3d.6 . 2 (𝜑𝐵𝑅𝐶)
51, 3, 4ertrd 8103 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4925   Er wer 8084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-er 8087
This theorem is referenced by:  nqereq  10153  efgred2  18651  xmetresbl  22765  pcophtb  23351  pi1xfr  23377  pi1xfrcnvlem  23378  erbr3b  30149  prtlem10  35483
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