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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 )