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Theorem ertrd 8741
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 8740 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 698 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5148   Er wer 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-co 5687  df-er 8725
This theorem is referenced by:  ertr2d  8742  ertr3d  8743  ertr4d  8744  erinxp  8810  nqereq  10959  adderpq  10980  mulerpq  10981  efgred2  19708  efgcpbllemb  19710  efgcpbl2  19712  pcophtb  24969  pi1xfr  24995  pi1xfrcnvlem  24996  erbr3b  32420  prjspner1  42050
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