Theorem eqvreltrd 39816

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 39815	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 712	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   class class class wbr 5106   EqvRel weqvrel 39324

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-refrel 39716  df-symrel 39748  df-trrel 39782  df-eqvrel 39793

This theorem is used by:  eqvreltr4d  39817