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Metamath Proof Explorer

Theorem ertr3d

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014)

Ref Expression
Hypotheses ersymb.1 ⊢ ( 𝜑 → 𝑅 Er 𝑋 )
ertr3d.5 ⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )
ertr3d.6 ⊢ ( 𝜑 → 𝐵 𝑅 𝐶 )
Assertion ertr3d ( 𝜑 → 𝐴 𝑅 𝐶 )

Proof

Step Hyp Ref Expression
1 ersymb.1 ⊢ ( 𝜑 → 𝑅 Er 𝑋 )
2 ertr3d.5 ⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )
3 ertr3d.6 ⊢ ( 𝜑 → 𝐵 𝑅 𝐶 )
4 1 2 ersym ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
5 1 4 3 ertrd ⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )