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Theorem ertr2d 8156
 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
ertr2d (𝜑𝐶𝑅𝐴)

Proof of Theorem ertr2d
StepHypRef Expression
1 ersymb.1 . 2 (𝜑𝑅 Er 𝑋)
2 ertrd.5 . . 3 (𝜑𝐴𝑅𝐵)
3 ertrd.6 . . 3 (𝜑𝐵𝑅𝐶)
41, 2, 3ertrd 8155 . 2 (𝜑𝐴𝑅𝐶)
51, 4ersym 8151 1 (𝜑𝐶𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   class class class wbr 4962   Er wer 8136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-er 8139 This theorem is referenced by:  pi1xfrcnvlem  23343
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