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Theorem eqvreltrd 38609
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvreltrd.1 (𝜑 → EqvRel 𝑅)
eqvreltrd.2 (𝜑𝐴𝑅𝐵)
eqvreltrd.3 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
eqvreltrd (𝜑𝐴𝑅𝐶)

Proof of Theorem eqvreltrd
StepHypRef Expression
1 eqvreltrd.2 . 2 (𝜑𝐴𝑅𝐵)
2 eqvreltrd.3 . 2 (𝜑𝐵𝑅𝐶)
3 eqvreltrd.1 . . 3 (𝜑 → EqvRel 𝑅)
43eqvreltr 38608 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 699 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5143   EqvRel weqvrel 38199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-refrel 38513  df-symrel 38545  df-trrel 38575  df-eqvrel 38586
This theorem is referenced by:  eqvreltr4d  38610
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