This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem eqvreltrd

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014) (Revised by Peter Mazsa, 2-Jun-2019)

Ref Expression
Hypotheses eqvreltrd.1 ⊢ φ → EqvRel R
eqvreltrd.2 ⊢ φ → A R B
eqvreltrd.3 ⊢ φ → B R C
Assertion eqvreltrd ⊢ φ → A R C

Proof

Step Hyp Ref Expression
1 eqvreltrd.1 ⊢ φ → EqvRel R
2 eqvreltrd.2 ⊢ φ → A R B
3 eqvreltrd.3 ⊢ φ → B R C
4 1 eqvreltr ⊢ φ → A R B ∧ B R C → A R C
5 2 3 4 mp2and ⊢ φ → A R C