Theorem eqvreltrd 38564

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 38563	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 698	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5166   EqvRel weqvrel 38152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-refrel 38468  df-symrel 38500  df-trrel 38530  df-eqvrel 38541

This theorem is referenced by:  eqvreltr4d  38565