Theorem tpeq1

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

	Ref	Expression
Assertion	tpeq1	⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐶 , 𝐷 } = { 𝐵 , 𝐶 , 𝐷 } )

Proof

Step	Hyp	Ref	Expression
1		preq1	⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } )
2	1	uneq1d	⊢ ( 𝐴 = 𝐵 → ( { 𝐴 , 𝐶 } ∪ { 𝐷 } ) = ( { 𝐵 , 𝐶 } ∪ { 𝐷 } ) )
3		df-tp	⊢ { 𝐴 , 𝐶 , 𝐷 } = ( { 𝐴 , 𝐶 } ∪ { 𝐷 } )
4		df-tp	⊢ { 𝐵 , 𝐶 , 𝐷 } = ( { 𝐵 , 𝐶 } ∪ { 𝐷 } )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐶 , 𝐷 } = { 𝐵 , 𝐶 , 𝐷 } )