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Theorem eqvreltr4d 35916
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvreltr4d.1 (𝜑 → EqvRel 𝑅)
eqvreltr4d.2 (𝜑𝐴𝑅𝐵)
eqvreltr4d.3 (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
eqvreltr4d (𝜑𝐴𝑅𝐶)

Proof of Theorem eqvreltr4d
StepHypRef Expression
1 eqvreltr4d.1 . 2 (𝜑 → EqvRel 𝑅)
2 eqvreltr4d.2 . 2 (𝜑𝐴𝑅𝐵)
3 eqvreltr4d.3 . . 3 (𝜑𝐶𝑅𝐵)
41, 3eqvrelsym 35912 . 2 (𝜑𝐵𝑅𝐶)
51, 2, 4eqvreltrd 35915 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5053   EqvRel weqvrel 35542
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-refrel 35824  df-symrel 35852  df-trrel 35882  df-eqvrel 35892
This theorem is referenced by:  eqvrelref  35917  eqvreldisj  35921
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