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Theorem ertr4d 8666
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 8659 . 2 (𝜑𝐵𝑅𝐶)
51, 2, 4ertrd 8663 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5105   Er wer 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-er 8647
This theorem is referenced by:  erref  8667  erdisj  8699  nqereu  10864  nqereq  10870  efgredeu  19532  pi1xfr  24416  pi1xfrcnvlem  24417  prjspner1  40942
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