Theorem tpeq3

Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011)

	Ref	Expression
Assertion	tpeq3	⊢ ( 𝐴 = 𝐵 → { 𝐶 , 𝐷 , 𝐴 } = { 𝐶 , 𝐷 , 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		sneq	⊢ ( 𝐴 = 𝐵 → { 𝐴 } = { 𝐵 } )
2	1	uneq2d	⊢ ( 𝐴 = 𝐵 → ( { 𝐶 , 𝐷 } ∪ { 𝐴 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝐵 } ) )
3		df-tp	⊢ { 𝐶 , 𝐷 , 𝐴 } = ( { 𝐶 , 𝐷 } ∪ { 𝐴 } )
4		df-tp	⊢ { 𝐶 , 𝐷 , 𝐵 } = ( { 𝐶 , 𝐷 } ∪ { 𝐵 } )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → { 𝐶 , 𝐷 , 𝐴 } = { 𝐶 , 𝐷 , 𝐵 } )