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Theorem eqvreltrd 36344
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 36343 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 699 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 5030   EqvRel weqvrel 35973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-refrel 36253  df-symrel 36281  df-trrel 36311  df-eqvrel 36321
This theorem is referenced by:  eqvreltr4d  36345
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