Theorem preq12d

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

	Ref	Expression
Hypotheses	preq1d.1	|- ( ph -> A = B )
	preq12d.2	|- ( ph -> C = D )
Assertion	preq12d	|- ( ph -> { A , C } = { B , D } )

Proof

Step	Hyp	Ref	Expression
1		preq1d.1	|- ( ph -> A = B )
2		preq12d.2	|- ( ph -> C = D )
3		preq12	|- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1 2 3	syl2anc	|- ( ph -> { A , C } = { B , D } )