Theorem preq12d

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

	Ref	Expression
Hypotheses	preq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	preq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	preq12d	⊢ ( 𝜑 → { 𝐴 , 𝐶 } = { 𝐵 , 𝐷 } )

Proof

Step	Hyp	Ref	Expression
1		preq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		preq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		preq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → { 𝐴 , 𝐶 } = { 𝐵 , 𝐷 } )
4	1 2 3	syl2anc	⊢ ( 𝜑 → { 𝐴 , 𝐶 } = { 𝐵 , 𝐷 } )