Theorem eqvreltrd 36721

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 36720	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 696	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5074   EqvRel weqvrel 36350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-refrel 36630  df-symrel 36658  df-trrel 36688  df-eqvrel 36698

This theorem is referenced by:  eqvreltr4d  36722