Theorem preq1

Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998)

	Ref	Expression
Assertion	preq1	⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } )

Proof

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