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Theorem ertr4d 8286
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 8279 . 2 (𝜑𝐵𝑅𝐶)
51, 2, 4ertrd 8283 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5042   Er wer 8264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-er 8267
This theorem is referenced by:  erref  8287  erdisj  8319  nqereu  10329  nqereq  10335  efgredeu  18857  pi1xfr  23639  pi1xfrcnvlem  23640
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