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Theorem ertrd 7803
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 7802 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 715 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4685   Er wer 7784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-co 5152  df-er 7787
This theorem is referenced by:  ertr2d  7804  ertr3d  7805  ertr4d  7806  erinxp  7864  nqereq  9795  adderpq  9816  mulerpq  9817  efgred2  18212  efgcpbllemb  18214  efgcpbl2  18216  pcophtb  22875  pi1xfr  22901  pi1xfrcnvlem  22902  erbr3b  29555
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