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Theorem ertrd 7743
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 7742 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 714 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4644   Er wer 7724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-co 5113  df-er 7727
This theorem is referenced by:  ertr2d  7744  ertr3d  7745  ertr4d  7746  erinxp  7806  nqereq  9742  adderpq  9763  mulerpq  9764  efgred2  18147  efgcpbllemb  18149  efgcpbl2  18151  pcophtb  22810  pi1xfr  22836  pi1xfrcnvlem  22837  erbr3b  29399
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