Theorem preq12

Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012)

	Ref	Expression
Assertion	preq12	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } )

Proof

Step	Hyp	Ref	Expression
1		preq1	⊢ ( 𝐴 = 𝐶 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐵 } )
2		preq2	⊢ ( 𝐵 = 𝐷 → { 𝐶 , 𝐵 } = { 𝐶 , 𝐷 } )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } )