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Theorem ertr4d 8001
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
ertr4d.5 (𝜑𝐴𝑅𝐵)
ertr4d.6 (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
ertr4d (𝜑𝐴𝑅𝐶)

Proof of Theorem ertr4d
StepHypRef Expression
1 ersymb.1 . 2 (𝜑𝑅 Er 𝑋)
2 ertr4d.5 . 2 (𝜑𝐴𝑅𝐵)
3 ertr4d.6 . . 3 (𝜑𝐶𝑅𝐵)
41, 3ersym 7994 . 2 (𝜑𝐵𝑅𝐶)
51, 2, 4ertrd 7998 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4843   Er wer 7979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-er 7982
This theorem is referenced by:  erref  8002  erdisj  8032  nqereu  10039  nqereq  10045  efgredeu  18480  pi1xfr  23182  pi1xfrcnvlem  23183
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