Theorem tppreq3

Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017)

	Ref	Expression
Assertion	tppreq3	⊢ ( 𝐵 = 𝐶 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		tpeq3	⊢ ( 𝐶 = 𝐵 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 , 𝐵 } )
2	1	eqcoms	⊢ ( 𝐵 = 𝐶 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 , 𝐵 } )
3		tpidm23	⊢ { 𝐴 , 𝐵 , 𝐵 } = { 𝐴 , 𝐵 }
4	2 3	eqtrdi	⊢ ( 𝐵 = 𝐶 → { 𝐴 , 𝐵 , 𝐶 } = { 𝐴 , 𝐵 } )