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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 )