Theorem eqvreltrd 38590

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	⊢ (𝜑 → 𝐴𝑅𝐵)
eqvreltrd.3	⊢ (𝜑 → 𝐵𝑅𝐶)

Assertion

Ref	Expression
eqvreltrd	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem eqvreltrd

Step	Hyp	Ref	Expression
1		eqvreltrd.2	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		eqvreltrd.3	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
3		eqvreltrd.1	. . 3 ⊢ (𝜑 → EqvRel 𝑅)
4	3	eqvreltr 38589	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 699	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5148   EqvRel weqvrel 38179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-refrel 38494  df-symrel 38526  df-trrel 38556  df-eqvrel 38567

This theorem is referenced by:  eqvreltr4d  38591