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 )