Theorem preq12i

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

	Ref	Expression
Hypotheses	preq1i.1	⊢ 𝐴 = 𝐵
	preq12i.2	⊢ 𝐶 = 𝐷
Assertion	preq12i	⊢ { 𝐴 , 𝐶 } = { 𝐵 , 𝐷 }

Proof

Step	Hyp	Ref	Expression
1		preq1i.1	⊢ 𝐴 = 𝐵
2		preq12i.2	⊢ 𝐶 = 𝐷
3		preq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → { 𝐴 , 𝐶 } = { 𝐵 , 𝐷 } )
4	1 2 3	mp2an	⊢ { 𝐴 , 𝐶 } = { 𝐵 , 𝐷 }