Theorem eqvreltrd 38592

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 38591	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 699	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5102   EqvRel weqvrel 38179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-refrel 38496  df-symrel 38528  df-trrel 38558  df-eqvrel 38569

This theorem is referenced by:  eqvreltr4d  38593