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Metamath Proof Explorer

Theorem eqvreltr4d

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

Ref Expression
Hypotheses eqvreltr4d.1 ⊢ φ → EqvRel R
eqvreltr4d.2 ⊢ φ → A R B
eqvreltr4d.3 ⊢ φ → C R B
Assertion eqvreltr4d ⊢ φ → A R C

Proof

Step Hyp Ref Expression
1 eqvreltr4d.1 ⊢ φ → EqvRel R
2 eqvreltr4d.2 ⊢ φ → A R B
3 eqvreltr4d.3 ⊢ φ → C R B
4 1 3 eqvrelsym ⊢ φ → B R C
5 1 2 4 eqvreltrd ⊢ φ → A R C